Mikayla Johnson in a group with Jordan Shields and Tayla Slay and Steven Williams

Chapter 8: Root and Radicals and Root Functions

Section 1: Radical expressions and Graphs

Section 2: Rational exponents

Section 3: Simplifying Radical expressions

Section 4: Adding and Subtracting Radical exponents

Section 5: Multiplying and Dividing Radius expression

Section 6: Solving Equation with Radius

Section 7: Complex Numbers

Chapter 1 list of topics:

Section 1.1

­Identities and contractions

­solving for a specified variable in literal equations

Section 1.2

­the additive property of inequality

­the multiplicative property of inequalities

­solving compound inequalities

Section 1.3

Solving absolute value equations

"Less than" absolute value inequalities

"Greater than" absolute value inequalities

Section 1.4

Indemnifying and simplifying imaginary and complex numbers

Adding and subtracting complex numbers

Division of complex numbers

Section 1.5

Quadratic equations and the zero product property

Quadratic equations and the square root property of equality

Solving quadratic equations by completing the square

Solving quadratic equations using the quadratic formula

Section 1.6

Polynomial equations of higher degree

Rational equations

Radical equations and equations with rational exponents

Equations in quadratic form

2.1: Rectangular Coordinates;

Graphing Circles,

Other Relation

2.2: Linear

Graphs

Rates of Change

2.3: Graphs

Special Forms of Linear Equations

2.4: Functions,

Function Notation,

the Graph of a Functions

2.5: Analyzing the graph of a function

2.6: Linear Functions

Real Data

Chapter 3

3.1 The toolbox function

transformation

3.2 Basic rational functions

power function

3.3 Variation:

The toolbox function

power function

3.4 Piecewise­Defined function

3.5 The algebra

Composition of functions

3.6 Another look at formulas,

functions,

problem solving

Section 1

Radian: Is the “a” √n a

Index: Is n √n a

Radical: Is the expression

Principal Root: is the notation √n a

Negative nTh Root:­√n a

Radical expression: Algebraic expression that contains radicals

Square root function:

Cube root function:

Section 3

Simplified: Has no factor raised to a power greater than or equal to index, radicand has no

fraction, no denominator contains radical

Pythagorean theorem: Relates the lengths of the tree sides or a right triangle

Hypotenuse: Longest side

Legs: Two Shorter sides

Section 4

Distance formula: d =√(x ) y − )2 − x12 + ( 2 y1

2

The root of the sum does not equal the sum of the root: =√9 6+ 1 / √9 + √16

Section 5

Conjugates: x=y and x­y are the congregates, example: is 11 + √2 − √2

Radical equation: An equation that includes one or more radical expressions with a variable

Section 6

Power rule: Raising both sides to a power

Extraneous solutions: Solutes that do not satisfy the original equation

Isolate radical: Make sure that one radical term is alone on one side of the equation

Apply power rule: Raise both sides of the equation to a power that is the same as the index of the

radical

Check: All proposed solutions in the original equation

Character independence: z in z = √l ÷ c

Section 7

Imaginary unit : Defined as i and i −i = √− 1 2 = 1

Complex number: ia = b

Real part: is a of ia = b

Imaginary part: b of ia = b

Pure Imaginary Number: and b =a = 0 / 0

Chapter 1

Section 1

Families of equations= group of equations that share common characteristics

Equation= a statement that 2 expressions are equal

Solutions/roots= replacement values that make the equation true

Equivalent equations= a sequence of simpler equations one simpler than the next next until it

reaches an obvious solution

Back­solution= checking a solution by plugging the solution back into the original problem

LCD= least common denominator

Contradiction equation= an equation that is true for one value, but false for another

Identity= equation that is always true

Contradictions= equations that are never true

Literal equation Do= one with two or more variables

Formula= models a known relationshi

Object variable= the variable that is being solved for

Section 1.2

Solution set= set of numbers that satisfy an inequality

Interval notation= symbolic way of indicating a selected interval of real numbers

Compound inequalities= applications of inequalities with more than one solution interval

Intersection= intersection of two sets A and B written A B­ set of elements common on both⋂

sides

Union= A B set of elements that are in either set (or both)⋃

Joint inequality= when the original inequality can be joined with another

Disjoint (disconnected) intervals= excluding a number or numbers

Non complex number: ia = b

Complex conjugates: i and a ia = b − b

Exact/closed formula= answers written using radicals

Quadratic formula= general solution used to solve equations belonging to the quadratic family

Definitions: For chapter 2

Section 1

­Relation: Correspondence between two sets

­Mapping Notation: Showing multiple corresponding values to another value

­Dependent Variable: The value that changes due to something

­Independent Variable: The value that changes by itself

­Domain: Set of all first coordinates

­Range: Set of all second coordinates

­Equation form: Another name for relations

­Tangular Coordinate System: Showing a relation on a graph

­X­axis: Horizontal lines on a graph

­Y­axis: Vertical lines on a graph

­XY­plane: Flat two dimensional surface

­Quadrants: Graph divided into four regions

­Grid Lines: Shown to denote integer values on a graph

­Coordinate Grid: Where the grid lines divide into

­Continuous: Graphs that go beyond forever

­Parabola: When x or y is squared

­Vertex: Lowest point on a graph

­Vertical Parabola: Parabola stretching upwards

­Semicircle: Half of a circle in two different quadrants

­Midpoint: Center between two endpoints

­Average Distance: Average between two endpoints

­Distance Formula: Find the distance between two points

­Radius: Fixed distances

­Center: Fixed point

­Central Circle: When both x and y coordinates of a circle are 0

­General Form: Basic form for an equation

Section 2

­ Y­Intercept: When X is zero and the Y has a value

­ X­Intercept: When Y is zero and the X has a value

­ Intercept Method: Graphing a linear equation using two points.

­ Slope of a Line: Rate of a line

­ Rate of Change: One quantity compared to another quantity in measurement

­ Delta: Represents change

­ Parallel Lines: Two lines in a plane that do not intersect

­ Perpendicular Lines: Two lines in a plane that intersect at right angles

Section 3

­ Secant Line: Straight line through two points on a non linear graph

­ Point­Slope Form: Isolating the Y value in a equation

Section 4

­ Function: Relation where each element of the domain corresponds to exactly one element of the

range

­ Vertex: The lowest value of the graph

­ Vertical Line Test: When a function has each X value on a vertical line once

­ Point of Inflection: Where the pivit point curves on a graph

­ Implied Domain: Set of all real numbers for which the function represents a real number

­ Function Notation: Used to express functions such as f(x)

Section 5

­ Even Functions: Symmetric about the Y­axis

­ Maximum Values: Y­Coordinate peaks from other graphs

­ Global Maximum: Largest y­value over the entire domain

­ Local Maximum: Largest range value in a specified interval

­ Endpoint Maximum: Endpoint of the domain

­ Odd Functions: Symmetric about the origin

­ End­Behavior: Describes the graph as it increases

­ Average Rate of Change: Slope of a secant line

Section 6

­ Scatterplot: Graph of all the order pairs in a data set

­ Positive Association: When data has larger input and output values

­ Negative Association: When data decreases left to right

Chapter 3

3.1: The Toolbox Functions and Transformations

­ The id function: f(x)= x

­ Square root function: f(x)= √x

­ Cubing function: f(x)= x^3

­ Squaring function: f(x)= x^2

­ Absolute value function: f(x)= x| |

­ Cube root: f(x)= √3 x

­ Toolbox Functions­gives us a variety of “tools” to model the real world.

­ Function Family­ set of functions defined by the same type of formula

­ Parent Function­ simplest function of a family of functions

­ Transformation­ when the parent graph may become “morphed” and/or shifted from

its original position, yet the graph will still retain its basic shape and features.

­ Axis of Symmetry­ a line of symmetry for a graph.

­ Vertical Shift/ Translation­ a translation in which the graph is moved up or down on the

y­axis

­ Horizontal Shift/ Translation­ a translation in which the graph is moved left or right on

the x­axis

­ Vertical Reflection­ translation in which the graph is reflected over the x­axis

­ Horizontal Reflection­ transformation in which the graph is reflected over the y­axis

­ Vertical Stretches­ transformation in which the graph is stretched due to the lead

coefficient being larger than one

­ Compression­ a transformation in which the graph is compressed due to the lead

coefficient being smaller than one

3.2: Basic Rational Function and Power Function

­ A rational function is one of the form V(x)= (where p and d are polynomials)d(x)p(x)

­ Rational function­ the ratio of two polynomials

­ Reciprocal square function­ are both above the x­axis

­ Asymptotic behavior­ becoming increasingly exact as a variable approaches a limit,

usually infinity

­ Horizontal asymptotic­ The line y=0 (the x­axis)

­ Vertical asymptote­ the line x=0 (the y=axis)

­ Power function­ where x is raised to some power

­ Allometric studies­ one area where power function and modeling with regression are

used extensively

3.3; Variation: The Toolbox Functions in Action

­ Direct Variation­ when the y­value varies directly with the x­value (as x increases, y

increases) y=kx

­ Constant of Variation­ in a direct variation is the constant (unchanged) ratio of two

variable quantities

­ Inverse Variation­ when the y­value varies indirectly with the x­value (as x increases, y

decreases) y=( )x1

­ Combined Variation­ when an equation involves both direct, and indirect variation.

­ Join Variation­ when one quantity is directly proportional to several others

­ Suppose a varies directly with b and inversely with c, find a when b=10 and c=5

­ The time it takes for a simple pendulum to complete one period (swing over and back)

is directly proportional to the square root of its length. If a pendulum 6 ft long has a

period of 3 seconds, find the time it takes for a 18­ft long pendulum to complete one

period.

­ A car’s speed varies directly with the radius of its tires and output (in rpm) of its

engine, but inversely with the product of the transmission and differential gear ratios.

If a car has a top speed of 50 mph at 5000 rpm in top gear in top gear(transmission

ratio of 1.0) what is its speed in first gear (transmission ratio of 2.5)

3.4: Piecewise­Defined Functions

­ Piecewise­defined function­ graphs may be various combinations of smooth/not

smooth and continuous/not continuous

­ Step function­ the pieces of the function form a series of horizontal step

­ Ceiling function­ is the largest integer not greater than x

­ Floor Function­ is the smallest integer not less than x

3.5: The Algebra and Composition of Functions

­ The notation used to represent the basic operation on two function is

­ (f + g)(x)= f(x) + g(x)

­ (f ­ g)(x)= f(x) ­ g(x)

­ (f g)(x)= f(x) g(x)∙ ∙

­ ( )(x)= ;g(x)fg

f(x)g(x) =/ 0

­ Algebra of Functions­ basic operations of functions

­ Composition of Functions­ when the input value is itself a function (rather than a single

number or variable)

Chapter 8 Equations

Section 1

Basic radical expression: (x) and f(x)f = √x = √3 x

Section 2

If is a real number then √n a a1/n = √n a

If m and n are positive integers with m/n in lowest terms then a )a−m/n = ( 1/n m

If (a = )am/n = 1am/n / 0

If all indicated roots are real numbers then a ) a )am/n = ( 1/n m = ( m 1/n

ar * as = ar+s a ) −r = ( 1ar as

ar = ar−s ( ) ba −r = ar

br (a ) r s = ars (ab) b r = ar r ( ) ba r = br

ar

)a−r = (a1 r

Section 3

If are real numbers an n is a natural number then and √n a √n b √n a * √n b = √n ab

ab)√n a * √n b = a1/n * b

1/n = ( 1/n = √n ab

If are real numbers and n is natural number then and √n a √n b =b / 0 √n ba * √n a

√n a

If m is an integer, n and k are natural numbers, and all indicated roots exist then √kn akm = √n am

c2 = a2 + b2

a= x|| 2 − x1||

Distance between and x , )( 2 y2 x , )( 2 y1

b = y|| 2 − y1||

From pythagrium theorem d2 = a2 + b2 d x ) y ) 2 = ( 2 − x12 + ( 2 − y1

Section 6

Radical equation x )( + y 2 = x2 + 2xy + y2

Section 7

√− b = i√b

Section formulas and equations for chapter 1:

Section 1.4

Standard form = (a+bi)

Complex conjugates= (a+bi) (a­bi)

Non complex number: ia = b

Complex conjugates: i and a ia = b − b

Section 1.5

Quadratic equations= a +bx+cx2

Square root property= = Kx2

Discriminant= ­4ac= determines the nature (real or imaginary) and the number of solutions tob2

a given quadratic equation

Chapter 2 Section 1

­Midpoint: ,( 2x₁+x₂

2y₁+y₂)

­ Distance Formula: ² √(x₂ ₁)− x +√(y₂ ₁)− y

­ Equation of a Circle: ²+(x )− h ²(y )− k ² = r

h=x­coordinate, k=y­coordinate, r=radius

Section 2

­ Linear Equations: ax+by=c

a, b, and c are real numbers

­ Slope Formula: m= x₂−x₁y₂−y₁

­ Horizontal Lines: y=k (0,k) is the y­intercept

­ Vertical Lines: x=h (h,0) is the intercept

Section 3

­ Slope Intercept Form: y=mx+b

m is the slope

y­intercept is (0,b)

­ Point Slope Form: y­y₁=m(x­x₁)

m is the slope

Section 5

­Even Functions: f(­x)=f(x)

­Odd Functions: f(­x)=­f(x)

­ Average Rate of Change: xy

­ Difference Quotient: ( hf(x+h)−f(x))

Problems

C0S4nc 1. Simplify

6√36 + 6

3√16 + 3√36

C0S7nc 2. Symplify

) − )(√− 6 + i2 + ( i2 + 3

C0S3nc 3. Symplify

43/12 * 48/12

C0S5nc 4. Find Z

h = √ z4l

C0S6nc 5. Simplify

√106592

CHAPTER 1:

All calculator problems. Simplify to the fullest, leave in exact form.

1.) 7x+20=18

2.) √x 5− 3 =

3.) ­3x­25 2≺ 4

4.) √− 26

5.) 3x x 2 − 7 − 6 = 0

Chapter 3

1. (C3S1nc) Identify the function family

a. f(x)= ­3(x+2)^2 ­5

b. f(x)= 21 x| + 4|

c. f(x)= √3 x − 4 + 2

d. f(x)= √3 x 6+ 1

2. (C3S1nc) Explain how the listed equations affect the function f(x)= (x)^2

a. f(x)= 3(x­10)^2

b. f(x)= (x+9)^2 ­ 361

c. f(x)= ­2(x+4)^2 + 6

3. (C3S1nc) Name The Parent function

Sketch the graph of each function, state transformations, label horizontal and vertical

asymptotes, and x and y intercepts, state domain and range

1. f(x)= +6x1

2. f(x)= 1x−6

3. f(x)= ­2x−1

4. f(x)= ­8−1(x−2)2

Answer for chapter 1 questions

1. .53

2. i√6 + 3

3. 411/12

4. z = 4lh2

5. 0, 591 6

1.(C3S3nc) Graph and name range and domain

2.(C3S4c) Choose one of the following graphs from the question above, and evaluate the

piecewise­defined function f(­3), f(4), and f(6)

(C3S5nc) For f(x)= x^2 +x and g(x) = 3x­2, find the following:

a. (f+g)(x)

b. (f­g)(2)

c. (f g)(3)∙

d. the domain of ( )(x)fg

2. (C3S5nc) For f(x)= x^2+ 3x and g(x)= 2x­9

a. (f*3)(g*6)

b. (f*g)(4)

c. (f+g)(3)