college algebra lecture notes

Copyright, MathDept,UIUC

COLLEGE ALGEBRA LECTURE NOTES©

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Date: Fall 2020. Contact: Alison Reddy, ared@illinois.edu.

0.1. Tentative Topic Schedule (Week/Topics).

(1) Real Numbers; Interval Notation; Absolute Value Integer Exponents; Radicals; Ra-

tional Exponents

(2) Algebraic Expressions; Factoring; Rational Expressions

(3) Equations: Basic, Quadratic, Other Types, Absolute Value; Modeling with Equations

(4) Inequalities: Linear, Non-linear, Absolute Value

(5) Coordinate Geometry; Graphing Equations; Symmetry; Lines

(6) Functions; Graphs of Function

(7) Increasing/Decreasing Functions; Avg Rate of Change; Transformations of Functions

(8) Quadratic Functions; Combining Functions

(9) One-to-One Functions and Inverses

(10) Polynomial Functions; Dividing Polynomials

(11) Real Zeros of Polynomials; Rational Functions

(12) Exponential Functions; Logarithmic Functions

(13) Laws of Logarithms; Exponential and Logaritmic Equations; Modeling with Exp and

Log Functions

(14) (Time Permitting) Unit Circle; Trig Functions of Real Numbers; Trig Graphs

1. Real Numbers

1.1. Properties of Real Numbers.

(1) Commutative Properties

a+ b = b+ a

ab = ba

(2) Associative Properties

(a+ b) + c = a+ (b+ c)

(ab)c = a(bc)

(3) Distributive Properties

a(b+ c) = ab+ ac

(b+ c)a = ba+ ca

(4) Additive Identity. The number 0 is the additive identity for any real number a.

a+ 0 = 0 + a = a

(5) Multiplicative Identity. The number 1 is the multiplicative identity for any real

number a.

a · 1 = a

(6) Inverse. Every nonzero real number a has an inverse, 1a, such that

a · 1

(7) Division. Division is equivalent to multiplication by the inverse

a÷ b = a · 1

where a is the numerator and b is the denominator.

(8) Properties of Fractions.

(a) ab· cd

= acbd

(b) ab÷ c

b· dc

= adbc

(c) ac

= a+bc

(d) ab

= adbd

+ cbdb

= ad+cbbd

(e) acbc

(f) If ab

= cd, then ad = bc.

1.2. Real Number Line. The real number line can be represented by points on a line.

The origin corresponds to the real number 0. The positive direction is towards the right

and negative direction is towards the left. Every real number is represented by a point on

the line and every point on the line corresponds to exactly one real number.

1.3. Inequalities. The real numbers are ordered. We use inequality symbols to indicate

how one real number relates to another.

(1) a < b means “a is less than b” and thus a lies to the left of b on the real number line.

(2) a > b means “a is greater than b” and thus a lies to the right of b on the real number

(3) a ≤ b means “a is less than or equal to b”.

(4) a ≥ b means “a is greater than or equal to b”.

1.4. Sets. A set is a collection of similar but distinct objects called elements. If S is a set,

the a ∈ S means a is an element of S; b /∈ S means b is not an element of S.

Example: E = {x|x is an even digit} = {0, 2, 4, 6, 8}

Example: A = {5, 6, 7}, B = {5, 6, 7, 8, 9}, C = {6, 7, 5}

Example: If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8}, and C = {8, 9, 10}, find

(1) A ∩B

(2) B ∪ C

(3) A ∩ C

1.5. Interval Notation.

Notation Set Description Graph

(a, b) {x|a < x < b}[a, b] {x|a ≤ x ≤ b}[a, b) {x|a ≤ x < b}(a, b] {x|a < x ≤ b}

(a,∞) {x|a < x}[a,∞) {x|a ≤ x}

(−∞, b) {x|x < b}(−∞, b] {x|x ≤ b}

(−∞,∞) R (set of all real numbers)

Example: Write each in interval notation and graph the interval.

(1) −2 ≤ x ≤ 0

(2) 2 < x < 7

(3) −52≤ x < 3

(4) x > −6

(5) x ≤ 2

Example: Write each interval as an inequality and graph the interval.

(1) (−5, 3)

(2) [6, 18]

(3) (−3.6, 8.4]

(4) (−∞, 5)

(5) [4,∞)

2. Exponents and Radicals

2.1. Exponential Notation. If a is any real number and n is a positive integer, then the

nth power of a is

an = a · a · . . . · a

The number a is called the base and n is called the exponent.

Example:

(1) −24

(2) (−2)4

2.2. Laws of Exponents. If a 6= 0 and n is a positive integer, then

(1) a0 = 1

(2) a−n =1

If a and b are any real numbers, and m and n are integers, then

(1) aman = am+n

an= am−n

(3) (am)n = amn

(4) (ab)n = anbn

(5)(ab

(6)(ab

)−n=

(7)a−m

b−n=

Example: Simplify and eliminate any negative exponent(s).

(1) 52 · (15)3

(2)3−2

(3)32 · 4−2 · 52−4 · 33 · 25

(4) (8x3)2

(5) (−3y)4

(6)(−2)3x4(yz)2

32xy3z

(3x−1

4y−1

2.3. Square Roots.√a = b means b2 = a and a ≥ 0.

Example:√

2.4. nth Roots. If n is a positive integer, then the principal nth root of a is defined as

n√a = b means bn = a

If n is even, then a ≥ 0 and b ≥ 0.

Example:

(1) 3√

(2) 4√−16

2.5. Properties of Radicals.

(1) n√ab = n

√a n√b

(3) m√

n√a = mn

(4) n√an = a if n is odd.

(5) n√an = |a| if n is even.

Example: Simplify each expression.

(1)√

√48√3

(3)4√

(4)√

3x2√

(5)√

32 +√

2.6. Rational Exponents. For rational exponents mn

in lowest terms, where m and n are

integers and n 6= 0, we define

amn = n

√am =

(n√a)m

If n is even, then a ≥ 0.

Example: Simplify each expression.

(1) 9− 32

(2) (16x2y−13 )

(3) (2√x)(3 3√x)

2.7. Rationalizing Denominators. Goal: To eliminate the radical in the denominator.

(1) If the denominator is of the form√a, then we multiply the numerator and the

denominator by√a and get

1√a·√a√a

√a√a2

(2) If the denominator is of the form n√am with m < n, then we multiply the numerator

and the denominator by n√an−m and get

1n√am·

n√an−m

=n√an−m

n√an−m+m

=n√an−m

n√an

=n√an−m

Example: Rationalize the denominator.

(1)−√

5√x3

6√x2

3. Algebraic Expressions

3.1. Algebraic Expressions. Combinations of variables (a letter that can represent any

number) and real numbers using the usual operations of addition, subtraction, multiplication,

and division.

Monomial: The terms of akxk of the polynomial.

Binomial: A combination of 2 monomials (terms).

3.2. Polynomials. Sum of monomials

anxn + an−1x

n−1 + . . .+ a1x1 + a0

an, an−1, . . . , a0 are real numbers. n is a nonnegative integer. If an 6= 0, then the polynomial

has degree n.

Example: Complete the table.

Type Degree Terms

x2 − 7x+ 3

3x5 + 5x

−4x3

3.3. Combining Polynomials.

(1) Adding and Subtracting Polynomials. We add and subtract polynomials by combin-

ing like terms. That is, terms with the same variables raised to the same power.

(2) Multiplying Algebraic Expressions. To find the product of polynomials or other

algebraic expressions, we repeatedly use the Distributive Property.

(3) Multiplying Two Binomials. FOIL. (First, Outer, Inner, Last)

Example.

(1) (x2 − 6x+ 5) + (2x− 3)

(2) 3(x3 − 2x2 + 1)− 5(2x3 − 4x2)

(3) −3x2(5x3 − 4)

(4) (x+ 2)(x2 − 3x+ 4)

3.4. Factoring. We use the Distributive Property to expand algebraic expressions. We

sometimes need to reverse this process by factoring an expression as a product of similar

(1) Factoring out common factors. Look for items common to every term.

(2) Factoring trinomials of the form x2 +Bx+C. Use trial and error to find two integers

whose product is C and whose sum is B.

x2 +Bx+ C = (x+ r)(x+ s)

Thus we need to choose numbers r and s such that r + s = B and r · s = C.

(3) Factoring trinomials of the form Ax2 +Bx+C. Use trial and error to find factors of

the form

Ax2 +Bx+ C = (px+ r)(qx+ s)

such that p · q = A, r · s = C, and ps+ qr = B.

Example. Factor completely.

(1) 3ax3 − 6ax2

(2) x2 + 7x+ 10

(3) x2 − 10x+ 16

(4) 2x2 + 5x+ 3

(5) 3x2 + 10x− 8

3.5. Special Factoring Formulas. A and B are algebraic expressions.

(1) Difference of Squares. A2 −B2 = (A−B)(A+B)

(2) Sum of Two Cubes. A3 +B3 = (A+B)(A2 − AB +B2)

(3) Difference of Two Cubes. A3 −B3 = (A−B)(A2 + AB +B2)

(4) Perfect Squares. (A+B)2 = A2 + 2AB +B2

(5) Factoring by Grouping. Polynomials with at least 4 terms can sometimes be factored

by grouping.

Example. Factor completely.

(1) 4x2 − 25

(2) 16x4 − 1

(3) x3 + 27

(4) x3 − 8

(5) x2 + 6x+ 9

(6) 4x2 − 12x+ 9

(7) x3 + x2 + 4x+ 4

(8) x−32 + 2x−

12 + x

4. Rational Expressions

4.1. Rational Expressions. A rational expression is a fractional expression where both the

numerator and the denominator are polynomials. The domain of an algebraic expression is

the set of real numbers that the variable is permitted to have.

Example. Find the domain of each expression.

x− 1

(x+ 1)2

(3)x3 − 2x

x2 − 5x+ 6

(4) 2x2 + 3x− 1

(6)√x+ 6

x2 − 5x+ 6

4.2. Simplifying Rational Expressions. We factor both the numerator and the denom-

inator and use the following property of fractions

This allows us to cancel out common factors.

Example.

(1)x3 − 2x2

x2 − 5x+ 6

(2)x2 − 2x

3x− 6

4.3. Multiplying and Dividing Rational Expressions.

(1) To multiply:A

B· CD

(2) To divide:ABCD

B· DC

Example.

(1)4x2

x2 − 16· x− 4

x2 + 7x+ 12

x2 − 7x+ 12x2 + x− 12

x2 − x− 12

Example. Factor each expression and simplify.

(1) (x− 3)(x+ 2) + 5(x− 3)(x+ 3)

(2) 4x√x+ 5− 9

√x+ 5

(3)x2(2x)− (x2 − 3)2x

(4) (x− 1)72 − (x− 1)

4.4. Adding and Subtracting Rational Expressions.

(1) First find a common denominator (or the least common denominator).

(2) UseA

Example.

(1)x+ 1

x− 3+

2x− 3

x− 3

x− 1− 2x− 3

(3)x+ 4

x2 − x− 2− 2x+ 3

x2 + 2x− 8

4.5. Compound Fractions. A compound fraction is a fraction in which the numerator,

the denominator, or both are themselves fractional expressions. To simplify a compound

fraction

(1) Combine the terms in the numerator into a single fraction.

(2) Combine the terms in the denominator into a single fraction.

(3) Invert and multiply.

Example.xy

1− yx

4.6. Rationalizing the denominator. If a fraction has a denominator of the form A +

B√C, we can rationalize the denominator by multiplying the numerator and the denom-

inator by the conjugate radical A − B√C. Note: (A + B

√C)(A − B

√C) = A2 − B2C.

Example. Rationalize the denominator.

2−√

Example. Rationalize the denominator.

2 +√

4.7. Avoiding Common Errors.

Correct Multiplication Property Common Error with Addition

(a · b)2 = a2 · b2 (a+ b)2 6= a2 + b2

√a · b =

√a√b, (a, b ≥ 0)

√a+ b 6=

√a+√b

√a2 · b2 = a · b, (a, b ≥ 0)

√a2 + b2 6= a+ b

a · b1

a−1 · b−1 = (a · b)−1 a−1 + b−1 6= (a+ b)−1

5. Equations

5.1. Solving Equations. The values of the unknown (variable) that make the equation

true are called the solutions or roots of the equation. The process of finding the solutions is

called solving the equation. Note: Two equations with exactly the same solutions are called

equivalent equations.

5.2. Properties of Equality.

(1) A = B ⇔ A+ C = B + C (Add the same thing to both sides.)

(2) A = B ⇔ cA = cB, c 6= 0 (Multiply both sides by the same nonzero number.)

5.3. Linear Equations. A linear equation in one variable is an equation equivalent to one

of the following form

ax+ b = 0

where a, b ∈ R and x is the variable.

Example. Solve the equation. 5 + 3x = 2x+ 7

5.4. Solving for a variable. Solve for one variable in terms of the others.

Example. Solve each equation for the indicated variable.

(1) a− 2[b− 3(c− x)] = 6, for x

(2)ax+ b

cx+ d= 2, for x

5.5. Modeling with Equations. Guidelines for Modeling with Equations.

(1) Identify the variable – what are you trying to find? Assign a variable to that quantity.

(2) Express the other unknown quantities in terms of the variable from step (1).

(3) Set up the model – find the relationship between the expressions from step (2) and

set up an equation.

(4) Solve the equation and check the answer.

Example. Solve the equation.

(1) Find 4 consecutive odd integers whose sum is 272. No trial-error answers will be

accepted.

(2) A woman earns 15% more than her husband. Together they make $69, 875 per year.

What is the husband’s annal salary?

(3) A rectangular garden is 10 feet longer than it is wide. Its area is 875 square feet.

What are its dimensions?

(4) What quantity of 60% acid solution must be mixed with 100 mL of 30% solution to

produce a 50% solution?

(5) A merchant blends a tea that sells for $2.90 per pound with a tea that sells for $2.65

per pound to produce 100 lbs of a mixture that sells for $2.80 per pound. How many

pounds of each type of tea does the merchant use in the blend?

(6) A water treatment plant uses batch deliveries of two levels of chlorine concentra-

tion to mix to create the level of concentration needed on any given day. Deter-

mine how many liters of a 3% chlorine solution and how many liters of a 5% chlo-

rine solution should be mixed to produce 500 L of a 4.4% chlorine solution. (Hint:

3%=0.03=3/100)

5.6. Quadratic Equations. A quadratic equation written in standard form, is an equation

of the form

ax2 + bx+ c = 0

where a, b, c ∈ R, a 6= 0, and x is the variable.

5.7. Zero Property of Real Numbers.

AB = 0⇔ A = 0 or B = 0

5.8. Solving Quadratic Equations by Factoring. Example. z2 = z + 6

5.9. Solving a Simple Quadratic Equation. The solutions of an equation of the form

x2 = c

are x =√c and x = −

√c, i.e. x = ±

Example. x2 = 7

Example. (x− 3)2 = 25

5.10. Solving Quadratic Equations by Completing the Square. The process of “ad-

justing” a quadratic equation to become a perfect square. To make x2 + bx a perfect square,

add(b2

)2to both sides of the equation. This gives

x2 + bx+

Example.

(1) x2 + 6x− 16 = 0

(2) x2 + 8x+ 7 = 0

(3) x2 − 4x− 23 = 0

5.11. The Quadratic Formula. The roots (zeros) if the quadratic equation

ax2 + bx+ c = 0

where a 6= 0, are

x =−b±

√b2 − 4ac

(1) 4x2 = 1− 2x

(2) 2x2 − 6x+ 7 = 0

5.12. The Discriminant. The discriminant of the general quadratic equation ax2+bx+c =

0, with a 6= 0 is

D = b2 − 4ac

(1) If D > 0, then the quadratic equation has 2 distinct real solutions.

(2) If D = 0, then the quadratic equation has exactly 1 real solution.

(3) If D < 0, then the quadratic equation has no real solutions.

Example. Use the discriminant to determine the number of real solutions of the equation.

4x2 +13

8= −5x

5.13. Solving an Equation that Involves Fractions. First multiply each side of the

equation by a common denominator (or LCD).

x− 12

x− 3+ 4 = 0

2(y − 3) =

x2 − 9=

x− 3

5.14. Equations Involving Radicals. The equation xn = a has the solutionx = n√a if n is odd

x = ± n√a if n is even and a ≥ 0

Note: It is possible to end up with one or more extraneous solutions so you must always check

your answers to make sure they satisfy the original equation.

Example. Find the real solutions of the equation.

(1) 3√

3x− 5− 3 = 0

(2)√x+ 4 = x+ 2

5.15. Equations of the Quadratic Type (Substitution). An equation of the form

aW 2 + bW + c = 0, where W is an algebraic expression, is an equation of the quadratic

(1) x4 − 5x2 + 4 = 0

(2) x13 + x

16 − 2 = 0

5.16. Modeling with Quadratic Equations. Solving word problems.

Example.

(1) A juggler tosses a ball into the air with a velocity of 40 ft/sec. from a height of 4 ft.

S = −16t2 + v0t+ s0

to find how long it takes for the ball to return to a height of 4 ft.

(2) A rectangular garden is 25 feet wide. If its area is 1125 square feet, what is the length

of the garden?

6. Inequalities

6.1. Rules for Inequalities. All of these rules also apply for the other inequality symbols:

A ≥ B, A < B, A > B.

(1) A ≤ B ⇔ A− C ≤ B − C(2) A ≤ B ⇔ A− C ≤ B − C(3) If c > 0, then A ≤ B ⇔ cA ≤ cB

(4) If c < 0, then A ≤ B ⇔ cA ≥ cB

(5) If A > 0 and B > 0, then A ≤ B ⇔ 1

A≥ 1

B(6) If A ≤ B and C ≤ D, then A+ C ≤ B +D

Example.

(1) Add 5 to each side of the inequality 4 < 8.

(2) Multiply each side of the inequality 4 < 8 by 2.

(3) Multiply each side of the inequality 7 > 5 by −3.

6.2. Linear Inequalities. An inequality is linear if the highest variable term is to the first

degree.

Example. Solve each inequality. Express the solution using interval notation and graph

the solution set.

(1) 3x− 1 ≥ 3 + x

(2) 4− 3(1− x) ≤ 3

(3) −5 < 4− 3x < 2

6.3. Absolute Value Equations. Recall, the absolute value of a, |a|, is the distance

from a to 0 on the real number line

a if a ≥ 0

-a if a < 0

Thus, |x| = c⇔ x = ±c.

(1) |x| = 6

(2) |2x+ 3| = 12

(3) 4− |2x| = 3

6.4. Absolute Value Inequalities.

Inequality Equivalent Form Graph

|x| < c −c < x < c

|x| ≤ c −c ≤ x ≤ c

|x| > c x < −c or x > c

|x| ≥ c x ≤ −c or x ≥ c

Example. Solve each inequality. Express the answer in interval notation.

(1) |x| < 5

(2) |x| > 5

(3) |3x+ 7| ≤ 5

(4) |1− 2x| > | − 3|

Example. A telephone company offers two long distance plans.

Plan A: $25 per month and 5¢per minute.

Plan B: $5 per month and 12¢per minute.

For how many minutes of long distance calls would plan B be financially advantageous?

6.5. Nonlinear Inequalities. Guidelines for Solving Nonlinear Inequalities.

(1) Move all terms to one side. If the non-zero side of the inequality involves quotients,

then put them together with a common denominator.

(2) Factor the non-zero side of the inequality.

(3) Find the intervals by determining the values for which each factor is zero. These

numbers will divide the real number line into intervals.

(4) Make a table or diagram and test a value within each interval to determine the sign

of the expression on that interval. Use this along with these rules –

(a) If a product/quotient has an EVEN number of negative terms, then the whole

expression is positive.

(b) If a product/quotient has an ODD number of negative terms, then the whole

expression is negative.

Example. Solve the non-linear inequality. Express the solution using interval notation and

graph the solution set.

(1) x2 − 3x− 10 < 0

(2) x2 − 5x+ 6 ≤ 0

(3) 2(x2 − 6) ≥ x(x+ 4)

7. Coordinate Geometry

7.1. Coordinate Plane. Ordered Pairs (x, y)

Example. Plot each point in a coordinate plane.

(2, 5), (−3,−6), (−4, 1), (1,−3), (0, 4), (−2, 0)

Example. Sketch the region given by each set.

(1) {(x, y) | y ≥ 0}

(2) {(x, y) | 1 ≤ x ≤ 2}

(3) {(x, y) | y = 3}

(4) {(x, y) | |x| > 4}

7.2. Absolute Value and Distance. The absolute value of a, |a|, is the distance from

a to 0 on the real number line

a if a ≥ 0

-a if a < 0

If a and b are real numbers, the distance between the points a and b on the real number

line is d(a, b) = |b− a|.

Example: Evaluate each expression.

(1) | − 6|

(2) |32|

(3) |10− π|

(4) −2− |2− | − 2||

7.3. Distance Formula. The distance between points A(x, y) and B(x, y) in the plane is

d(A,B) =√

(x2 − x1)2 + (y2 − y1)2

Example. Find the length of the line segment between the points (−3, 7) and (2, 3).

Example. Which point C(−6, 3) or D(3, 0) is closer to point E(−2, 1)?

7.4. Midpoint Formula. The midpoint of a line segment from A(x1, y1) to B(x2, y2) is

(x1 + x2

2,y1 + y2

Example. Find the midpoint of the line segment between the points (−3, 7) and (2, 3).

7.5. Graphs of Equations in Two Variables. x is the independent variable; y is the

dependent variable. Pick value(s) for x and solve for y.

Example. Sketch the graph of the equation 3x+ y = 4.

Example. Sketch the graph of the equation y = x2 + 2.

7.6. Intercepts.

(1) x-intercepts: The x-coordinate of the points where the graph intersects the x-axis.

Find them by setting y = 0 and solving for x.

(2) y-intercepts: The y-coordinate of the points where the graph intersects the y-axis.

Find them by setting x = 0 and solving for y.

Example. Find the intercepts of the graph of the equation y = 4x+ 7.

Example. Find the intercepts of the graph of the equation y = x2 + 5x+ 6.

7.7. Symmetry.

(1) Symmetric with respect to the x-axis. Graph does not change when reflected in the

x-axis. Equation is equivalent when y is replaced with −y.

(2) Symmetric with respect to the y-axis. Graph does not change when reflected in the

y-axis. Equation is equivalent when x is replaced with −x.

(3) Symmetric with respect to the origin. Graph does not change when rotated 180o

about the origin. Equation is equivalent when x is replaced with −x; and y is

replaced with −y.

Example. Test the equation for symmetry and sketch the graph.

(1) y = x2 + 8

(2) y2 = x− 4

(3) y = −7x

8. Linear Equations

8.1. The Slope of a Line. The slope m of a non-vertical line that passes through the

points A(x1, y1) and B(x2, y2) is

m =∆y

∆x=y2 − y1x2 − x1

Note: The slope of a vertical line is undefined.

Example. Find the slope of the line that passes through the points P (1, 2) and Q(3, 3).

8.2. Point-Slope Form of the Equation of a Line. An equation of the line that passes

through the point (x1, y1) and has slope m is

y − y1 = m(x− x1)

where (x, y) is fixed, and (x1, y1) and m are given.

Example. Find an equation of the line through (−1, 3) with slope m = 6.

Example. Find an equation of the line throughs (3, 2) and (−3, 4).

8.3. Slope-Intercept Form of the Equation of a Line. An equation of the line that has

slope m and y-intercept b is

y = mx+ b

Example. Find an equation of the line with slope m = 3 and y-intercept b = −2.

Example. Find the slope m and the y-intercept b of the line 3x+ 5y = 6.

8.4. Vertical Lines. An equation of a vertical line through (a, b) is

8.5. Horizontal Lines. An equation of a horizontal line through (a, b) is

8.6. General Equation of a Line. The graph of every linear equation Ax + By + C = 0

with A and B not both zero is a line. Conversely, every line is the graph of a linear equation.

Example. Find the slope and intercepts of the line −3x− 5y+ 30 = 0 and draw its graph.

8.7. Parallel Lines. Two non-vertical lines are parallel if and only if they have the same

slope. Any two distinct vertical lines are parallel.

8.8. Perpendicular Lines. Two lines with slopes m1 and m2 are perpendicular if and only

if m1m2 = −1. That is, the slopes are negative reciprocals, m2 =−1

and m1 =−1

A horizontal line (m = 0) is perpendicular to a vertical line (m undefined).

Example. Find an equation of the line through the point (2, 5) that is parallel to the line

6x+ 4y + 5 = 0.

Example. Find and equation of the line perpendicular to the line x + 5y = 8 that passes

through the point (3,−4).

8.9. Slope as a Rate of Change. When a line is used to model the relationship between

two quantities, the slope of the line is the rate of change of one quantity with respect to the

other.

Example. Some scientists believe that the average surface temperature of the world has

been rising steadily. The average surface temperature is given by

T = 0.02t+ 8.50

where T is the temperature in oCelcius and t is years since 1900.

(1) What do the slope and T -intercept represent? Draw and label the axes.

(2) Use the equation to predict the average global surface temperature in 2100.

9. Functions

9.1. Definition of a Function. A function f is a rule that assigns to each element x in a

set A exactly one element, f(x), in a set B.

Example. Determine if each relationship is a function or not.

(1) {(1, 1), (2, 4), (3, 9)}

(2) {(1, 5), (2, 6), (3, 7), (4, 5)}

(3) {(1, x), (1, y), (2, y), (3, z)}

9.2. Function Notation. We refer to f(x) as the value of f at the number x. If y = f(x),

then x is the independent variable and y is the dependent variable or the value of f at x.

9.3. Evaluating a Function. Example. If f(x) = 3x2 − 2x, evaluate the following:

(1) f(5)

(2) f(a) + f(b)

(3) f(a+ b)

(4) f(x− 1)

Example. Given the piecewise defined function.

f(x) =

x2 if x < 0

x+ 1 if x ≥ 0

(1) Evaluate the function at the given values.

(a) f(−2)

(b) f(0)

(c) f(2)

(2) Graph the function.

9.4. Domain of a Function. The domain is the set of all real numbers for which the

expression is defined as a real number. For each x in the domain of a function f , there is

one and only one image f(x) in the range.

Example. Determine the domain of each function.

(1) f(x) = x3 + 2x

(2) g(x) =6x

x2 − 9

(3) h(x) =√

9− 2x

(4) k(x) =

√x− 2√4− x

9.5. Four Ways to Represent a Function.

(1) Verbally: By a description in words.

(2) Algebraically: By an explicit formula.

(3) Visually: By a graph.

(4) Numerically: By a table of values.

10. Graphs of Functions

10.1. The Graph of a Function. If f is a function with domain A, then the graph of f is

the set of ordered pairs

{(x, f(x)) | x ∈ A}

In other words, the graph of f is the set of all points (x, y) such that y = f(x); that is, the

graph of f is the graph of the equation y = f(x).

10.2. Determining Information from the Graph of a Function. Example. Consider

the function f with the following graph.

−2 2

(1) What is the domain of f? [Answer in interval notation.]

(2) What is the range of f? [Answer in interval notation.]

(3) Find f(1) and f(−2).

(4) If f(x) = 3, what is x?

(5) Does the graph display symmetry? If so, which type?52

10.3. Graphing Piecewise Defined Functions. A piecewise defined function is defined

by different equations on different parts of its domain. The graph of such a function consists

of separate pieces.

Example. Sketch the graph of the piecewise defined function.

f(x) =

1 if x ≤ 1

x+1 if x > 1

10.4. Vertical Line Test. A curve in the coordinate plane is the graph of a function if and

only if no vertical line intersects the curve more than once.

Example. Use the vertical line test to determine which are graphs of functions.

10.5. Graphs that Define Functions. Any equation in the variables x and y defines a

relationship between those two variables. An equation defines a function if it gives exactly

one value of y for each value of x. Note, not every equation defines y as a function of x.

Example. Determine whether or not each equation defines y as a function of x.

(1) x2 + 2y = 4

(2) x = y2

10.6. Linear Functions. f(x) = mx+ b

Example.

(1) f(x) = 4

(2) f(x) = 3x+ 2

10.7. Power Functions. f(x) = xn

Example.

(1) f(x) = x2

(2) f(x) = x3

(3) f(x) = x4

10.8. Root Functions. f(x) = n√x

Example.

(1) f(x) =√x

(2) f(x) = 3√x

(3) f(x) = 4√x

10.9. Reciprocal Functions. f(x) =1

Example.

(1) f(x) =1

(2) f(x) =1

10.10. Absolute Value Function.

f(x) = |x| =

x if x ≥ 0

-x if x < 0

11. Increasing and Decreasing Functions; Average Rate of Change

11.1. Increasing and Decreasing Functions.

(1) f is increasing on an interval I if f(x1) < f(x2) whenever x1 < x2 in I.

(2) f is decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.

y = f(x)

Example. Determine the following about the function from the graph. On its domain:

(−4, 2)

(0,−2) (1,−3)

(6, 2)

(1) Where is it increasing?

(2) Where is it decreasing?

(3) Where is it constant?

11.2. Average Rate of Change. The average rate of change of the function y = f(x)

between x = a and x = b is

ARC =∆y

∆x=f(b)− f(a)

b− a

The average rate of change is the slope of the secant line between x = a and x = b on

the graph of f , that is, the slope of the line that passes through the points (a, f(a)) and

(b, f(b)).

y = f(x)

b− a

f(b)f(b)− f(a)

Example. Find the average rate of change of f(x) = 5x3 from

(1) 1 to 6

(2) 0 to x

Example. Changing water levels. The graph shows the depth of the water W in a reservoir

over a one-year period as a function of the number of days x since the beginning of the year.

(0, 50)

(100, 75) (150, 85)

(200, 50)(300, 25)

(1) Determine the intervals on which the function W is increasing; decreasing.

(2) What is the average rate of change between x = 100 and x = 200?

11.3. Average Rate of Change of Linear Functions. Example. Calculate the average

rate of change between several points for y = f(x) = −2x+ 5.

What can we conclude about the average rate of change of a linear function?

12. Transformations of Functions

12.1. Vertical and Horizontal Shifts. Suppose c > 0. To graph

(1) y = f(x) + c shift the graph of y = f(x) up c units.

(2) y = f(x)− c shift the graph of y = f(x) down c units.

(3) y = f(x− c) shift the graph of y = f(x) right c units.

(4) y = f(x+ c) shift the graph of y = f(x) left c units.

Example. Graph each function.

(1) f(x) = x2

(2) g(x) = x2 + 4

(3) h(x) = x2 − 3

(4) j(x) = (x− 2)2

(5) k(x) = (x+ 4)2

12.2. Reflections about the x-axis and y-axis. To graph

(1) y = −f(x), reflect the graph of y = f(x) about the x-axis.

(2) y = f(−x), reflect the graph of y = f(x) about the y-axis.

(1) f(x) =√x

(2) f(x) = −√x

(3) f(x) =√−x

Example. Determine the function that is graphed after the following transformations are

applied to the graph of y = x2. Graph the function.

(1) Reflect about the x-axis.

(2) Shift up 3 units.

(3) Shift left 4 units.

Example. Use transformations to graph the function f(x) = −(x− 1)3 − 1

12.3. Vertical Stretching and Shrinking of Graphs. To graph y = cf(x)

(1) If c > 1, stretch the graph of y = f(x) vertically by c units.

(2) If 0 < c < 1, shrink the graph of y = f(x) vertically by c units.

12.4. Horizontal Stretching and Shrinking of Graphs. To graph y = f(cx)

(1) If c > 1, shrink the graph of y = f(x) horizontally by 1c

units.

(2) If 0 < c < 1, stretch the graph of y = f(x) horizontally by 1c

units.

(1) f(x) = |x|

(2) g(x) = 3|x|

(3) h(x) =1

(4) k(x) = |2x|

(5) j(x) = |12x|

12.5. Even and Odd Functions. Let f be a function.

(1) f is even if f(−x) = f(x) for all x in the domain of f .

(2) f is odd if f(−x) = −f(x) for all x in the domain of f .

Note: Even functions are symmetric with respect to the y-axis; odd functions are sym-

metric with respect to the origin.

Example. Determine, by the graph, if the function is even, odd, or neither.

Example. Determine if each function is even, odd, or neither.

(1) f(x) = x2 − 3

(2) g(x) = 4x3 − 3x

13. Quadratic Functions

13.1. Graphing Quadratic Functions. A quadratic function, f(x) = ax2 + bx + c, can

be converted to standard form, f(x) = a(x− h)2 + k by completing the square. In either

form, the graph of f(x) is a parabola with vertex (h, k) that opens up if a > 0 and opens

down if a < 0. The vertex is the maximum or minimum point of the parabola.

13.2. Properties of the Graph of the Quadratic Function.

f(x) = ax2 + bx+ c, with a 6= 0

(1) Vertex = (h, k) =

(−b2a, f

(−b2a

))(2) Axis of symmetry: The line x = −b

(3) Opens up if a > 0, vertex is a minimum.

(4) Open down if a < 0, vertex is a maximum.

(5) If b2 − 4ac > 0, then the graph has 2 distinct x-intercepts.

(6) If b2− 4ac = 0, then the graph has exactly 1 x-intercept and it touches the x-axis at

its vertex.

(7) If b2 − 4ac < 0, then the graph has no x-intercepts and does not cross nor touch the

x-axis.

(8) y-intercept at f(0)

13.3. Maximum or Minimum Values of a Quadratic Function. The maximum or

minimum value of a quadratic function

f(x) = ax2 + bx+ c, with a 6= 0

occurs at x =−b2a

(1) If a > 0, then the minimum value is f(−b2a

)(2) If a < 0, then the maximum value is f

(−b2a

Example. Graph the function. Determine its maximum or minimum value.

(1) f(x) = x2 + 2x− 8

(2) f(x) = −2x2 + 12x− 18

(3) f(x) = x2 + 2x− 1

(4) f(x) = −2x2 + 4x− 3

13.4. Modeling with Functions. Guidelines.

(1) Express the Model in Words. Identify the quantity you want to model and express

it, in words, as a function of the other quantities in the problem.

(2) Choose the Variable. Identify all the variables used to express the function in step

(1). Assign a variable, such as x, to one quantity and express the other values in

terms of this variable.

(3) Set up the Model. Express the model in algebraic terms by writing it as a function

of the single variable chosen in step (2).

(4) Use the Model. Use the function to answer the question posed in the problem.

Example. A rectangle has a perimeter of 20 feet. Find a function that models its area A

in terms of the length x of one of its sides.

Example. Find two numbers whose sum is 24 and whose product is a maximum.

Example. A farmer with 4000 meters of fencing wants to enclose a rectangular plot that

borders on a river. If the farmer does not fence the side along the river, what is the largest

area that can be enclosed?

Example. A ball is thrown vertically upward from the top of a building 96 feet tall with

an initial velocity of 80 feet per second. The distance s (in feet) of the the ball from the

ground after t seconds is

s = 96 + 80t− 16t2

(1) After how many seconds does the ball strike the ground?

(2) After how many seconds will the ball pass the top of the building in its way down?

Example. The monthly revenue R achieved by selling x wristwatches is figured to be

R(x) = 75x− 0.2x2. The monthly cost C of selling x wristwatches is C(x) = 32x+ 1750.

(1) How many wristwatches must the company sell to maximize revenue? What is the

maximum revenue?

(2) Profit is defined as P (x) = R(x)− C(x). What is the profit function?

(3) How many wristwatches must the company sell to maximize profit? What is the

maximum profit?

14. Combining Functions

14.1. Algebra of Functions. Let f and g be functions with domains A and B, then

(1) Sum: (f + g)(x) = f(x) + g(x)

(2) Difference: (f − g)(x) = f(x)− g(x)

(3) Product: (f · g)(x) = f(x) · g(x)

(4) Quotient:

)(x) =

g(x), g(x) 6= 0

Example. Let f(x) = x2 + 3x and g(x) = 2x− 1. Find the following and their domains.

(1) (f + g)(x)

(2) (f − g)(x)

(3) (f · g)(x)

14.2. Composition of Functions. Given 2 functions f and g, the composite function,

(f ◦ g)(x), also called the composition of f and g, is defined by

(f ◦ g)(x) = f(g(x))

The domain of (f ◦ g)(x) is the set of all numbers in the domain of g such that g(x) is in the

domain of f .

Example. Let f(x) = 3x3 + 4x2 − 5 and g(x) = 2x. Find the following.

(1) (f ◦ g)(1)

(2) (g ◦ f)(1)

(3) (f ◦ f)(−3)

(4) (f ◦ g)(x)

(5) (g ◦ f)(x)

Example. Let f(x) =1

x+ 3and g(x) =

x− 2. Find (f ◦ g)(x) and determine its domain.

Example. Find functions f and g such that (f ◦ g)(x) = H, if

(1) H = (2x+ 3)4

(2) H =√x2 + 1

15. One-to-One Functions and their Inverses

15.1. One-to-One Functions. A function with domain A is called one-to-one function,

1-1, if no 2 elements in A have the same image, that is

f(x1) 6= f(x2) whenever x1 6= x2

In other words, if f(x1) = f(x2) then x1 = x2. A function that is either increasing or

decreasing over its domain is 1-1.

15.2. Horizontal Line Test. A function is 1-1 if and only if no horizontal line intersects

its graph more than once.

Example. Determine, by the graph, if the function is 1-1.

15.3. The Inverse of a Function. Let f be a 1-1 function with domain A and range B,

then its inverse function, f−1 has domain B and range A, and is defined by

f−1(y) = x⇐⇒ f(x) = y

for any y ∈ B. Note: a function is 1-1 if and only if it has an inverse function.

Example. Find the inverse of the function.

(1) {(−3,−27), ((−2,−8), ((−1,−1), (0, 0), (1, 1), (2, 8), (3, 27)}

15.4. Property of Inverse Functions. Let f be a 1-1 function with domain A and range

B. The inverse function f−1 satisfies the following properties:

(1) f−1(f(x)) = x

(2) f(f−1(x)) = x

Example. Verify that f(x) = 2x+ 3 and f−1(x) =1

2(x− 3) are inverse functions.

15.5. How to Find the Inverse of a 1-1 Function.

(1) Write y = f(x).

(2) Interchange x and y.

(3) Solve for y in terms of x (if possible).

(4) The resulting equation is the inverse: y = f−1(x).

15.6. Geometric Interpretation. The graph of f−1 is obtained by reflecting the graph of

f over the line y = x.

Example.

(1) Find the inverse of the function f(x) = 4x− 8

(2) Determine the domain and range of f and f−1.

(3) Graph f and f−1 on the same axes.

Example. f(x) =3x+ 2

x− 1is a 1-1 function.

(1) Find the inverse of f(x).

(2) Determine the domain and range of f and f−1.

16. Polynomial Functions and their Graphs

16.1. Polynomial Functions. A polynomial function of degree n is a function of the form

P (x) = anxn + an−1x

n−1 + . . .+ a1x1 + a0

where n is a non-negative integer and an 6= 0. The numbers an, an−1, . . . , a1, a0 are called

the coefficients of the polynomial. The number an, the coefficient of the highest power, is

the leading coefficient and the term anxn is the leading term. The number a0 is the constant

coefficient or constant term. The graph of a polynomial function is a smooth curve – no

breaks and no sharp corners.

16.2. End Behavior. The polynomial P (x) = anxn + an−1x

n−1 + . . . + a1x1 + a0 has the

same end behavior as the monomial Q(x) = anxn, so its end behavior is determined by the

degree n and the sign of the leading coefficient an.

(1) If y = P (x) has odd degree and

(a) the leading coefficient is positive, the y → ∞ as x → ∞ and y → −∞ as

x→ −∞.

(b) the leading coefficient is negative, the y → ∞ as x → −∞ and y → −∞ as

x→∞.

(2) If y = P (x) has even degree and

(a) the leading coefficient is positive, the y → ∞ as x → ∞ and y → ∞ as

x→ −∞.

(b) the leading coefficient is negative, the y → −∞ as x→ −∞ and y → −∞ as

x→∞.

Example. Determine the end behavior of the polynomial function.

(1) f(x) = 4x5 − 24x4 + 52x3 − 56x2 + 48x− 32

(2) f(x) = 4(x2 + 1)(x− 2)3

16.3. Real Zeros of Polynomials. If P is a polynomial function and c is a real number,

then the following are equivalent:

(1) c is a zero of P .

(2) x = c is a solution of the equation P (x) = 0.

(3) x− c is a factor of P .

(4) (c, 0) is an x-intercept of the graph of P .

Example. Form a polynomial function whose zeros are −3, 0, 4 and whose degree is 3.

16.4. Multiplicity. If the factor (x − c) appears m times in the complete factorization of

P (x), then we say that c is a zero of multiplicity m.

(1) If c is a zero of even multiplicity, then the graph touches the x-axis at c.

(2) If c is a zero of odd multiplicity, then the graph crosses the x-axis at c.

Example. Form a polynomial function of degree 3 whose zeros are −1 with multiplicity 1

and 3 with multiplicity 2.

16.5. Intermediate Value Theorem. If P (x) is a polynomial function and P (a) and P (b)

have opposite signs, then there exists at least one value c between a and b for which P (c) = 0.

16.6. Local Extrema of Polynomials. If P (x) = anxn + an−1x

n−1 + . . . + a1x1 + a0 is a

polynomial of degree n, then the graph of P has at most n− 1 local extrema.

16.7. Guidelines for Graphing Polynomial Functions.

(1) Zeros. Factor the polynomial to find all its real zeros; these are the x-intercepts of

the graph.

(2) Test Points. Make a table of values for the polynomial. Include test points to

determine whether the graph of the polynomial lies above or below the x-axis on the

intervals determined by the zeros. Include the y-intercept in the table.

(3) End behavior. Determine the end behavior of the polynomial.

(4) Multiplicity. Determine where the graph crosses or touches the x-axis at each x-

intercept.

(5) Graph. Plot the intercepts and other points you found. Sketch a smooth curve that

passes through these points and exhibits the correct end behavior.

Example. For each polynomial function:

(1) Determine the degree and the end behavior.

(2) Determine the number of local extrema.

(3) Find the x and y intercepts.

(4) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(5) Sketch the graph.

(a) P (x) = x3 − 2x2 − 3x

(b) P (x) = x3(x+ 2)(x− 3)2

(c) P (x) = x2(x− 2)(x2 + 3)

(d) P (x) = −2x2(x− 3)(x+ 6)

(e) P (x) = 4(x2 + 1)(x− 2)3

17. Dividing Polynomials

17.1. Long Division. Quotient x Divisor + Remainder = Dividend.

Example. Divide 409 by 12

Example. Divide 4x3 − 3x2 + x+ 1 by x− 4

17.2. Division Algorithm. If P (x) and D(x) are polynomials with D(x) 6= 0, then there

exists unique polynomials Q(x) and R(x) such that

P (x) = Q(x) ·D(x) +R(x)

R(x) is either 0 or of degree less than the degree of D(x).

Example. Divide 1− x2 + x4 by x2 + x+ 1

17.3. Synthetic Division. Use synthetic division when the divisor is of the form x− c.

Example. Use synthetic division to find the quotient Q(x) and the remainder R(x) when

f(x) is divided by g(x).

(1) f(x) = 3x3 + 2x2 − x+ 3; g(x) = x− 4

(2) f(x) = x5 − 4x3 + x; g(x) = x+ 3

Question: What do we mean when we say that 3 is a factor of 51?

Example. Use synthetic division to determine whether x− 2 is a factor of

f(x) = 3x4 − 6x3 − 5x+ 10

17.4. Remainder and Factor Theorems.

(1) If a polynomial P (x) is divided by x− c, then the remainder is the value P (c).

(2) c is a zero of P (x) if and only if x − c is a factor of P (x). That is, x − c is a factor

of P (x) if and only if P (c) = 0.

Example. Determine the remainder if f(x) = x3 − 6x2 − 3 is divided by x+ 3.

Example. Use the Factor Theorem to determine whether x + 2 is a factor of f(x) =

3x6 + 2x3 − 176.

18. Real Zeros of Polynomials

18.1. Rational Zeros Theorem. If the polynomial P (x) = anxn+an−1x

n−1+. . .+a1x1+a0

has integer coefficients, then every rational zero of P is of the form pq

where p is a factor of

the constant coefficient a0 and q is a factor of the leading coefficient an.

Example. List all the possible rational zeros of P (x) = 2x3 − 13x2 + 24x− 9

Note: The maximum number of real zeros is the degree of the polynomial.

18.2. Finding the Real Zeros of a Polynomial.

(1) List Possible Rational Zeros. List all the possible rational zeros using the Rational

Zeros Theorem.

(2) Test. Use the Factor Theorem to test the possible rational zeros you found in step

(1). When you find a zero go to step (3).

(3) Divide. Use synthetic division to evaluate the polynomial at the rational zero you

found in step (2). Note the quotient you have obtained.

(4) Repeat. Repeat the above steps for the quotient. Stop when you reach a quotient

that is quadratic or factors easily and use the quotient formula or factor to find the

remaining zeros.90

Example. Find the real zeros, write f in factored form, and graph the function.

(1) f(x) = 2x3 − 13x2 + 24x− 9

(2) f(x) = 2x3 + 11x2 − 7x− 6

18.3. The Upper and Lower Bounds Theorem. Let P be a polynomial with real coef-

ficients.

(1) If we divide P (x) by x− b (with b > 0) using synthetic division and if the row that

contains the quotient and the remainder has no negative entry, then b is an upper

bound for the real zeros of P .

(2) If we divide P (x) by x− a (with a < 0) using synthetic division and if the row that

contains the quotient and the remainder has entries that are alternately nonposi-

tive and nonnegative, then a is an lower bound for the real zeros of P .

Note: The phrase “alternately nonpositive and nonnegative” simply means that the signs of

the numbers alternate, with 0 considered to be positive or negative as required.

Example. Show that the given values for a and b are lower and upper bounds for the real

zeros of the polynomial. P (x) = 2x3 + 5x2 + x− 2; a = −3, b = 1

Example. Find the real zeros of the polynomial and graph the function.

(1) P (x) = x3 − 3x2 + 4

19. Rational Functions

19.1. Rational Functions. A rational function is a function of the form R(x) =P (x)

Q(x)where P (x) and Q(x) are polynomials, Q(x) 6= 0 and P (x) and Q(x) have no common

factors.

Example. Find the domain of each rational function.

(1) f(x) =3x4 − 7

(2) g(x) =x3 − 1

x2 + 1

(3) h(x) =x

x2 − 9

19.2. Definition of Vertical and Horizontal Asymptotes.

(1) The line x = a is a vertical asymptote of the function y = f(x) if y approaches

±∞ as x approaches a from the right or left.

(a) y →∞ as x→ a+

(b) y →∞ as x→ a−

(c) y → −∞ as x→ a+

(d) y → −∞ as x→ a+

(2) The line y = b is a horizontal asymptote of the function y = f(x) if y approaches

b as x approaches ±∞.

(a) y → b as x→∞(b) y → b as x→ −∞

19.3. Asymptotes of Rational Functions. Let r be the rational function

r(x) =P (x)

anxn + an−1x

n−1 + . . .+ a1x1 + a0

bmxm + bm−1xm−1 + . . .+ b1x1 + b0

(1) The vertical asymptotes of r are the lines x = a, where a is a zero of the denominator.

(2) If n < m, then r has a horizontal asymptote y = 0.

(3) If n = m, then r has a horizontal asymptote y =anbm

(4) If n > m, then r has no horizontal asymptote.

Example. Find the asymptotes for each rational function.

(1) f(x) =x

(2) g(x) =2x+ 3

x2 + 6x− 16

(3) k(x) =6x2 − x+ 1

2x2 − 8

19.4. Slant (Oblique) Asymptotes and End Behavior. If r(x) = P (x)Q(x)

is a rational func-

tion in which the degree of the numerator is greater than the degree of the denominator, we

can use the Division Algorithm to express the function in the form

r(x) = ax+ b+R(x)

where the degree of R is less than the degree of Q and a 6= 0. This means that as x→ ±∞,R(x)Q(x)→ 0. Thus for large values of |x|, the graph of y = r(x) approaches the graph of the

line y = ax+ b. In this situation we say that y = ax+ b is a slant or oblique asymptote.

19.5. Analyzing the Graph of a Rational Function.

(1) Find the domain of the rational function.

(2) Write R in lowest terms.

(3) Locate the x-intercepts of the graph. The x-intercepts, if any, of

R(x) =p(x)

in lowest terms satisfy the equation p(x) = 0.

(4) Locate the y-intercepts of the graph. The y-intercept, if there is one, is R(0).

(5) Locate the vertical asymptotes. The vertical asymptotes, if any, of

R(x) =p(x)

in lowest terms are found by identifying he real zeros of q(x). Each zero of the

denominator gives rise to a vertical asymptote.

(6) Locate the horizontal or oblique asymptotes, if any, using the procedure given above.

Determine the points, if any, at which the graph of R(x) intersects these asymptotes

[solve R(x) = asymptote].

(7) List the zeros of the numerator and the denominator of R. Create a table to locate

points on the graph around each of these zeros.

(8) Use the above results to graph the function.

Example. Graph the rational function.

(1) R(x) =3

x2 − 4

(2) R(x) =3x+ 6

x2 + 2x− 8

(3) R(x) =x2 + 2x− 8

x2 − x− 6

(4) R(x) =x2 − 2x− 8

(5) R(x) =x2 − 3x− 4

20. Exponential Functions

20.1. Exponential Functions. The exponential function with base a is define for all real

numbers x by

f(x) = ax

where a > 0 and a 6= 1.

Example. Let f(x) = 5x. Evaluate the following.

(1) f(2)

(2) f(−23)

(3) f(π)

(4) f(√

20.2. Graphs of Exponential Functions. The exponential function f(x) = ax, with a > 0

and a 6= 1, has domain (−∞,∞) and range (0,∞). The graph of f has one of the following

two shapes.

Example. Find the domain, range, and the asymptote for each function. Sketch the graph.

(1) f(x) = (15)x

(2) f(x) = 5x

(3) f(x) = 5x + 2

20.3. The Natural Exponential Function. The natural exponential function is the ex-

ponential function

f(x) = ex

with base e. It is referred to as the exponential function.

Example. Sketch the graph of the function.

(1) f(x) = ex

(2) f(x) = e(x+3)

21. Logarithmic Functions

21.1. Logarithmic Functions. Let a be a positive number with a 6= 1. The logarithmic function

with base a, where a > 0 and a 6= 1, is denoted by logax and is defined by

logax = y ⇔ ay = x

Note, the logarithmic and exponential functions are inverses.

21.2. Converting between Exponential and Logarithmic Expressions. Example. Change

each expression to an equivalent expression involving a logarithm or exponential.

(1) 3.44 = v

(2) ex = 12

(3) b5 = 26

(4) logb6 = 8

(5) logey = −2

(6) log5d = 3

Example. Find the exact value.

(1) y = log327

(2) y = log2116

21.3. Properties of Logarithms (and their exponential equivalents).

(1) loga1 = 0

a0 = 1

(2) logaa = 1

a1 = a

(3) logaax = x

ax = ax

(4) alogax = x

logax = logax

Example. Evaluate each expression.

(1) 3log37

(2) log636

(3) log31

(4) log10√

(5) log44

21.4. Graphs of Logarithms. The logarithm function f(x) = logax, a > 0 and a 6= 1, has

domain (0,∞) and range (−∞,∞). The graph of f has one of the following two shapes.

21.5. Finding the Domain of a Logarithmic Function. As the logarithmic and expo-

nential are inverses functions, y = ax is the inverse function of y = logax, the domain of

the logarithmic function equals the range of the exponential function and the range of the

logarithmic function equals the domain of the exponential function. That is,

Domain Log Fcn = Range Exp Fcn = (0,∞)

Range Log Fcn = Domain Exp Fcn = (−∞,∞)

Example. Find the domain, range, and the asymptote for each function. Sketch the graph.

(1) f(x) = log3(x+ 2)

(2) h(x) = log5x+ 2

(3) g(x) = ex + 2

21.6. Common Logarithm Function. The logarithm with base 10 is called the common

logarithm and is denoted by omitting the base.

log10x = logx

Example. Graph f(x) = log(x− 2). Determine the domain, range, and vertical asymptote.

21.7. Natural Logarithm Function. The logarithm with base e is called the natural

logarithm and is denoted by ln.

logex = lnx

Example. Graph f(x) = ln(x+ 3). Determine the domain, range, and vertical asymptote.

21.8. Properties of Natural Logarithms (and their exponential equivalents).

(1) ln1 = 0

e0 = 1

(2) lne = 1

e1 = e

(3) lnex = x

ex = ex

(4) elnx = x

lnx = lnx

Example. Evaluate.

(1) lne7

(2) ln

(3) eln2

22. Properties of Logarithms

22.1. Properties of Logarithms. Let a > 0 and a 6= 1 and let A,B,C be any real numbers

with A > 0 and B > 0, then

(1) loga(AB) = logaA+ logaB

(2) logaA

B= logaA− logaB

(3) loga(AC) = C · logaA

Example. Simplify and evaluate, if possible, each expression.

(1) log337

(2) log48 + log42

(3) log2− log5

(4) log(xyz

22.2. Expanding and Combining Logarithmic Expressions. Use the above Properties

of Logarithms to expand or combine logarithms with the same base.

Example.

(1) ln5x√

1− 3x

(x− 4)3

(2) log8 + logx+ log(x2 + 3)− log7

23. Exponential and Logarithmic Equations

23.1. Exponential Equations. An exponential equation is one in which the variable occurs

in the exponent. For example, 4x = 16.

Example. Solve the equation. 9x − 3x − 72 = 0

23.2. Guidelines for Solving Exponential Equations.

(1) Isolate the exponential expression on the left hand side of the equation.

(2) Take the logarithm of each side and use the Properties of Logarithms to “bring down

the exponent”.

(3) Solve for the variable.

Example. Solve the equation. 3 · 4x = 18

Example. Solve the equation. 3x−5 = 25x+3

Example. Solve the equation. x2ex + xex − 2ex = 0

23.3. Logarithmic Equations. A logarithmic equation is one in which a logarithm of the

variable occurs. For example, log2(x+ 2) = 5.

23.4. Guidelines for Solving Logarithmic Equations.

(1) Isolate the logarithmic term on the left hand side of the equation; you may first need

to combine the logarithmic terms.

(2) Write the equation in exponential form.

(3) Solve for the variable.

(4) Remember that the domain of the log function is (0,∞).

(1) logx+ log(x− 3) = 1

(2) log4(x+ 1)− log4x = 2

(3) log8(x− 4) + log8(3 + x) = 1

(4) 2log7x = log716

(1) 102x = 5

(2) e1−4x = 2

(3) 5x = 4x+1

(1) lnx = 10

(2) log23 + log2x = log25 + log2(x− 2)

(3) log9(x− 5) + log9(x+ 3) = 1

23.5. Compound Interest. Compound interest is calculated by the formula

A(t) = P(

)ntWhere

(1) A(t) is the amount after t years

(2) P is the principal amount

(3) r is the interest rate per year (in decimal notation)

(4) n is the number of times the interest is compounded per year

(5) t is the number of years

Example. Suppose you invest $1500 at an annual rate of 5% compounded quarterly. How

much money will you have after 1 year? What if the investment was compounded monthly?

23.6. Continuously Compounded Interest. Continuously compounded interest is cal-

culated by the formula

A(t) = Pert

(1) A(t) is the amount after t years

(2) P is the principal amount

(3) r is the interest rate per year (in decimal notation)

(4) t is the number of years

Example. Suppose you invest $1500 at an annual rate of 5% compounded continuously.

How much money will you have after 1 year? Two and a half years?

Example. Lily has $200 to invest at 7% per year. If the investment is compounded

continuously,

(1) How much money will she have after 2 year?

(2) How long will it be before she has $300?

(3) How long will it be before she double her investment?

(4) Does the amount of the initial investment effect the doubling time?

24. Modeling with Exponential and Logarithmic Equations

24.1. Exponential Growth Model. A population that experiences exponential growth

increases according to the model

n(t) = n0ert

(1) n(t) is the population at time t

(2) n0 is the initial size of the population

(3) r is the relative rate of growth (in decimal notation)

(4) t is the time

Example. The size P of a certain insect population at time t (in days) obeys the function

P (t) = 750e0.04t

(1) Determine the number of insects at t = 0 days.

(2) What is the growth rate of the insect population?

(3) What is the population after 10 days?

(4) When will the insect population reach 800?

(5) When will the insect population double?

24.2. Radioactive Decay Model. If m0 is the initial mass of a radioactive substance with

half-life h, then the mass remaining at time t is modeled by the function

m(t) = m0e−rt

where r = ln2h

or h = ln2r

Example. A certain radioactive material decays according to the function

A(t) = A0e−0.012t

where A0 is the initial amount present and A is the amount present at time t (in years).

Suppose a scientist has a sample of 650g of the material.

(1) What is the decay rate of the material

(2) How much of the material will be left after 10 years?

(3) When will 400g of the material be left?

(4) What is the half-life of this material?

Example. The population of a colony of mosquitoes grows exponentially. If there are 1000

mosquitoes initially and there are 1800 after 1 day, what is the size of the colony after 3

days? When will the population reach 10, 000 mosquitoes?

Example. The half-life of radium is 1690 years. If 10 grams are present now, how much

will be present in 50 years?

25. Final Review

25.1. This is a good start, but is not a complete review.

(1) Solve. −2(5− 3x) + 8 = 4 + 5x

(2) Multiply the polynomials using the FOIL method, express your answer as a single

polynomial in standard form. (x+ 3)(x+ 5)

(3) Factor. x2 + 5x+ 6

(4) Solve. 4√

(5) Use synthetic division to determine whether x− 2 is a factor of 4x3 − 3x2 − 8x+ 4.

(6) Reduce to lowest terms.3x+ 9

x2 − 9

(7) Solve. Express your answer in interval notation and graph the solution set.

3x− 1 ≥ 3 + x

(8) Given the points: (-1, 2); (5, -2)

(a) Find the distance between the points.

(b) Find the midpoint of the line segment joining the points.

(c) Find the slope.

(9) Answer the questions about the given function. f(x) =x+ 2

x− 6(a) Is the point (3,14) on the graph of f?

(b) If x = 4, what is f(x)?

(c) If f(x) = 2, what is x?

(d) What is the domain of f?

(e) List the x -intercepts, if any, of the graph of f .

(f) List the y - intercept, if there is one, of the graph of f.

(10) Given the following linear function. y = −3x+ 4

(a) Determine the slope and y-intercept of the function.

(b) Graph the linear function. [Label at least 2 points.]

(c) Determine whether the linear function is increasing, decreasing, or constant.

(d) Find an equation for the line with slope m = 3 and containing the point (-2,3).

Write your answer in slope-intercept form.

(11) Blending Teas. The manager of a store that specializes in selling tea decides to

experiment with a new blend. She will mix some Earl Grey tea that sells for $5 per

pound with some Orange Pekoe tea that sells for $3 per pound to get 100 pounds of

the new blend. The selling price of the new blend is to be $4.50 per pound, and there

is to be no difference in revenue from selling the new blend versus selling the other

types. How many pounds of the Earl Grey tea and Orange Pekoe tea are required?

(12) Solve the inequality. Express your answer in interval notation and graph the solution

set. |3x− 2| ≤ 4

(13) Given the quadratic function: f(x) = x2 + 6x+ 9

(a) Use the quadratic formula to find the real solution(s).

(b) Find the vertex and the axis of symmetry.

(c) Find the y-intercept.

(d) Graph the function.

(14) Use transformations to graph the function. f(x) = (x − 1)3 + 2 [Label the point of

inflection.]

(15) Given the function f(x) = (x− 2)(x+ 4)2

(a) List each real zero and its multiplicity, and determine whether the graph crosses

or touches the x-axis at each x-intercept.

(b) Describe the end behavior or find the power function that the graph of f resembles

for large values of |x|.

(c) Graph the function.

(16) Use the Factor Theorem to determine whether x-c is a factor of f . f(x) = 2x3 +

8x2 − 5x+ 5; c = 2

(17) f(x) = 2x8 − x7 + 8x4 − 2x3 + x+ 3

(a) Give the maximum number of real zeros that the polynomial function may have.

(b) List the potential rational zeros of the polynomial function. [Do not attempt to

find the zeros.]

(18) The function f(x) = 4x+ 2 is one-to-one.

(a) Find its inverse.

(b) Check that f−1(f(x)) = x.

(c) Check that f(f−1(x)) = x.

(d) Graph f and f−1 on the same coordinate axes.

(19) Analyze the graph of the function R(x) =2x− 6

x(a) Find the domain of the rational function.

(b) Find the x-intercept.

(c) Find the vertical asymptote.

(d) Find the horizontal or oblique asymptote.

(e) Graph the function.

(20) (a) Change the exponential expression to an equivalent logarithmic expression.

52 = z

(b) Change the logarithmic expression to an equivalent exponential expression.

log5u = 13

(c) Write as the sum and/or difference of logarithms. log3xy2

(21) Amplifying Sound. An amplifier’s power output P (in watts) is related to its decibel

voltage gain d by the formula

P (d) = 25e0.1d

(a) Find the power output for a decibel voltage gain of 4 decibels. [Write an equa-

tion, but do not evaluate.]

(b) For a power output of 50 watts, what is the decibel voltage gain? [i.e. At what

value of d, will the power output double?]

(22) Solve the non-linear inequality. 2x3 + 5x2 − 3x > 0

(23) Solve the equation.x+ 5

x− 2=

x2 − 4

(24) Solve the equation.√

2x+ 1 + 1 = x

(25) Solve the equation x2 + 6x − 2 = 0 (a) by completing the square and (b) by the

quadratic formula.

(26) Simplify the expression.

12 b−3)4

(3a4b−32 )2

26. Trigonometry

26.1. The Unit Circle. The unit circle is the circle of radius 1 centered at the origin in

the xy-plane. Its equation is

x2 + y2 = 1

Example.

(1) Show that the point

13,−12

)is on the unit circle.

(2) The point P is on the circle. Given that the y-coordinate of P is−1

3and P is in

quadrant IV, find P (x, y).

26.2. Terminal Points. Suppose t is a real number. Mark off a distance t along the unit

circle, starting at the point (1, 0) and moving in a counterclockwise direction if t is positive

or in a clockwise direction if t is negative. In this way we arrive at a point P (x, y) on the

unit circle. The point P (x, y) obtained this way is called the terminal point determined by

the real number t. A full rotation is 2π.

Example. Find the terminal point P (x, y) on the unit circle determined by the given value

(1) t =3π

(2) t =−π2

Example. Suppose that the terminal point determined by t is the point

)on the

unit circle. Find the terminal point determined by each of the following.

(1) π − t

(2) −t

(3) π + t

(4) 2π + t

26.3. Reference Number. Let tbe a real number. The reference number t̄ associated with

t is the shortest distance along the unit circle between the terminal point determined by t

and the x-axis. [Note: The reference number is always positive.]

Example. Find the reference number for each value of t.

(1) t =5π

(2) t =7π

(3) t = −4π

(4) t =π

26.4. Using Reference Numbers To Find Terminal Points. To find the terminal point

P determined by any value of t, we use the following steps:

(1) Find the reference number t̄.

(2) Find the terminal point Q(a, b) determined by t̄.

(3) The terminal point determined by t is P (±a,±b), where the signs are chosen accord-

ing to the quadrant in which this terminal point lies.

Example. Find (a) the reference number for each value of t, and (b) the terminal point

determined by t.

(1) t =2π

(2) t =13π

(3) t = −11π

27. Trigonometric Functions of Real Numbers

27.1. Definition of the Trigonometric Functions. Let t be any real number and let

P (x, y) be the terminal point on the unit circle determined by t. We define

(1) sin t = y

(2) cos t = x

(3) tan t = xy, x 6= 0

(4) csc t = 1y, y 6= 0

(5) sec t = 1x, x 6= 0

(6) cot t = xy, y 6= 0

Note: Because the trigonometric functions can be defined in terms of the unit circle, they

are sometimes called the circular functions.

Example. Find the six trigonometric functions for t =π

27.2. Domains of the Trigonometric Functions.

Function Domain

sin, cos All real numbers

tan, sec All real numbers except π2

+ nπ for any integer n

cot, csc All real numbers except nπ for any integer n

27.3. Signs of the Trigonometric Functions.

Quadrant Positive Functions Negative Functions

I all none

II sin, csc cos, sec, tan, cot

III tan, cot sin, csc, cos, sec

IV cos, sec sin, csc, tan, cot

Example. Find the exact value of the trigonometric function at the given real number.

(1) sin7π

(2) cos8π

(3) tanπ

(4) cotπ

(5) secπ

(6) csc−3π

Example. The terminal point P (x, y) determined by t is given. Find sin t, cos t, and tan t.

3,−2√

Example. Find the sign of the expression, tan t sec t, if the terminal point determined by is

in quadrant IV.

Example. Find the quadrant in which the terminal point t lies if, tan t > 0 and sin t < 0.

27.4. Even – Odd Property. Sine, cosecant, tangent, and cotangent are odd functions;

sine and secant are even functions.

(1) sin (-t) = -sin t

(2) cos (-t) = cos t

(3) tan (-t) = -tan t

(4) csc (-t) = -csc t

(5) sec (-t) = sec t

(6) cot (-t) = -cot t

Example. Determine whether the function is even, odd, or neither.

(1) f(x) = x2 sinx

(2) f)x) = sinx cosx

(3) f(x) = x3 + cosx

27.5. Reciprocal Identities.

(1) csc t = 1sin t

(2) sec t = 1cos t

(3) cot t = 1tan t

(4) tan t = sin tcos t

(5) cot t = cos tsin t

27.6. Pythagorean Identities.

(1) sin2 t+ cos2 t = 1

(2) tan2 t+ 1 = sec2 t

(3) 1 + cot2 t = csc2 t

Example. Find the value of the trigonometric functions of t if

(1) cos t = −4

5and the terminal point of t is in quadrant II

(2) tan t = −3

4and cos t > 0

28. Trigonometric Graphs

28.1. Graphs of the Sine and Cosine Functions. The sine and cosine functions are

periodic according to the following definition: A function f is periodic if there is a positive

number p such that f(t+ p) = f(t) for every t. The least such positive number (if it exists)

is the period of f . If f has period p, then the graph of f on any interval of length p is called

one complete period of f .

28.2. Periodic Properties of Sine and Cosine. The sine function has period 2π: sin(t+

2π) = sin t. The cosine function has period 2π: cos(t+ 2π) = cos t.

28.3. Graphs of the Sine and Cosine Functions. The graph of the sine function is

symmetric with respect to the origin; sine is an odd function. The graph of the cosine

function is symmetric with respect to the y-axis; cosine is an even function.

Example. Graph the following functions.

(1) y = sinx

(2) y = − sinx

(3) y = cosx

(4) y = − cosx

(5) y = 1 + sinx

(6) y = 1− cosx

In general, for the functions y = a sinx and y = a cosx, the number |a| is called the amplitude

and is the largest value these functions attain.

28.4. Sine and Cosine Curves. The sine and cosine curves

y = a sin kx

y = a cos kx

have amplitude|a| and period2π

An appropriate interval on which to graph one complete period is

Example. Find the amplitude and the period of the function, and graph one complete

period.

(1) y = 3 sin 3x

(2) y = cos(10πx)

28.5. Shifted Sine and Cosine Curves. The sine and cosine curves

y = a sin k(x− b)

y = a cos k(x− b)

have amplitude|a|, period2π

k, and phase shift b.

An appropriate interval on which to graph one complete period is

[b, b+

Example. Find the amplitude, period and phase shift of the function, and graph one

complete period.

(1) y = cos(x− π2)

(2) y = −2 sin(x− π6)

Example. Variable Stars are ones whose brightness varies periodically. One of the most

visible is R Leonis; its brightness is modeled by the function

b(t) = 7.9− 2.1 cos( π

where t is measured in days.

(1) Find the period of R Leonis.

(2) Find the maximum and minimum brightness.

Example. The graph of one complete period of a sine or cosine curve is given.

(1) Find the amplitude, period, and phase shift.

(2) Write an equation that represents the curve in the form y = a sin(kx).

(3) Write an equation that represents the curve in the form y = a cos k(x− b).

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