math 121. practice problems from chapter 2 fall...

Math 121. Practice Problems from Chapter 2 Fall 2016

Chapter 2: Key types of Problems

Section 1

1. Cartesian coordinate system. For practice see Exercises 3, 5.

2. Midpoint of two points. For practice see Exercises 1, 2, 4.

3. Distance between points. For practice see Exercises 6, 7

4. Equations of circles, center and radius of a circle. For practice see Exercises 8, 9, 10.

Section 2

1. Interpret graph of function. For practice see Exercises 1.

2. Evaluation of functions from definition. For practice see Exercises 2, 3.

3. Find domain of a function. For practice see Exercise 4.

4. Find zeros of functions. For practice see Exercises 5, 6.

5. Graph a function by plotting points. For practice see Exercises 7, 8.

6. Determine where a function is increasing, decreasing or constant. For practice see Exer-cises 9, 10.

7. Recognize functions and one-to-one functions from graphs. For practice see Exercise 11.

8. Create a function from a description (word problem). For practice see Exercise 12.

Section 3

1. Determine properties and equations of lines from their graphs. For practice see Exer-cises 1, 2.

2. Slope and equation of line through two points. For practice see Exercises 3 and 2.

3. Find equation of line parallel to a given line. For practice see Exercise 6.

4. Find equation of line perpendicular to a given line. For practice see Exercises 4, 5.

5. Find equation of line with given slope through a point. For practice see Exercise 5.

6. Word problems involving finding and using linear modesl. For practice see Exercises 7,8, 9.

Section 4

1. Determine properties (max/min, range, axis of symmetry, vertex) and equations ofquadratic functions from their graphs. For practice see Exercises 1, 2.

2. Determine properties (max/min, range, axis of symmetry, vertex) and equations ofquadratic functions given their equations. For practice see Exercises 3, 4.

3. Find the vertex of a quadratic functions given its equation. For practice see Exercises 5,6.

4. Applications of quadratic functions. Projectile motion; see Exercise 7. Geometry, seeExercies 8 and 9.

Section 5

1. Determine whether a function is even, odd or neither. For practice see Exercises 1, 2.

2. Determine whether a graph of an equation possesses symmetry over the x-axis, y-axis,origin. For practice see Exercise 3.

3. Horizontal and vertical translations of functions. For practice see Exercises 4, 5, 6, 7, 8,9.

4. Reflections of graphs over x-axis or y-axis. For practice see Exercise 10.

5. Horizontal and vertical stretching or shrinking. For practice see Exercise 11

6. Combinations of translations, reflections, stretchings. For practice see Exercises 12, 13,14, 15, 16.

Section 6

1. Addition, subtraction, multiplication and division of function, For practice, see Exer-cises 1, 2.

2. Finding compositions of functions. For practice see Exercises 3, 4.

3. Evaluate algebraic combinations or compositions of functions. For practice see Exercise 5.

4. Find and simplify a difference quotient of a function. For practice see Exercises 6, 7.

5. Application of functions. For practice see Exercise 8.

6. Define a piecewise function from its graph. For practice see Exercise 9.

1 Cartesian Coordinates

1. Find the midpoint of the line segment with the given endpoints.

(−1, 8), (−7, 4).

2. Find the other endpoint of the line segment that has the given endpoint and midpoint.

Endpoint (−6, 2), Midpoint (−4, 1).

3. Plot the following points on the Cartesian plane

(5, 4), (−5,−5), (5,−2), (−5, 2)

4. Find the midpoint of the given points

(−7,−4) and (5,−6)

and then plot the two points, the line segment between them, and the midpoint on agraph.

5. Four points A, B, C and D are plotted below. Find their coordinates.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

A

BC

D

x

y

6. Find the distance between the given points. (You are not required to simplify the answer)

(5, 0) and (−3, 5)

7. The points(−6, 4) and (3,−4)

are plotted as the end points of the hypotenuse of a right-triangle with sides labeled aand b.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

(−6, 4)

(3,−4)

cb

a

x

y

(a) Find the length of the sides a and b.

(b) Use the Pythagorean theorem to find c2 where c is the length of the hypotenuse.

(c) Let d be the distance between the points (−6, 4) and (3,−4). Use the distance formulato find d2.

8. A circle has a diameter with endpoints (2, 7) and (−8, 5). Find the equation of the circlein standard form. (Hint: the center is the midpoint of the endpoints of the diameter).

9. Write the equation of the circle x2 + y2− 10x+ 8y+ 40 = 0 in standard form. Then findthe center and radius of the circle.

10. Write the equation of the circle x2 + y2 + 6x− 10y+ 25 = 0 in standard form. Then findthe center and radius of the circle.

2 Introduction to Functions

1. Use the graph of the function f given below to answer the following questions.

(a) Find the x-intercept(s), if any, of the graph of f .

(b) Find the y-intercept(s), if any, of the graph of f .

(c) Find f(−4)

(d) Find f(−2)

(e) Find f(2)

(f) Find f(4)

(g) Find f(6)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

2. Let f(z) = 2z2 − 4z. Find the following

(a) f(−5) (b) f(5) (c) f(5 + h) (d) f(x+ h)

3. Consider the piecewise defined function f(x) =

3− 3x, if x ≤ −4;

5 if − 4 < x < 4;

x2 − 5 if x ≥ 4.

Find:

(a) f(−5) (b) f(−4) (c) f(−3) (d) f(4) (e) f(t+ 4) for t ≥ 0

4. Determine the domains of the following functions.

(a) f(x) =(x+ 4)(x+ 7)√

4− x2.

(b) g(x) =√x− 4

(c) h(x) =√

4− x

(d) k(x) = − 2√4− x

(e) r(x) =3(x− 4)

(x+ 4)(x− 2)

5. Find the zeros of the function f given below; that is find all x so that f(x) = 0.

f(x) = x4 + 13x3 + 42x2.

6. Find the zeros of the function f given below; that is find all x so that f(x) = 0.

f(x) = 8x3 + 5x2 − 16x− 10.

7. Let y = 4− |x+ 2|.(a) Complete the following table.

x −6 −5 −4 -3 -2 -1 0 1 2

y

(b) Plot the graph of y = 4− |x+ 2| and determine its x-intercepts and y-intercepts.

8. (a) Find the domain of the function

f(x) =√x+ 6− 1

(b) Sketch the graph of f(x) =√x+ 6− 1 by plotting appropriate points.

9. Determine intervals on which the function graphed below is: (a) increasing; (b) decreas-ing; (c) constant.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

10. Determine intervals on which the function graphed below is: (a) increasing; (b) decreas-ing; (c) constant.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

11. Two graphs are given below. For each graph, determine whether it is a graph of afunction, and if it is, determine whether the function is be one-to-one. Explain youranswers.

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

12. An open box is to be made from a square piece of cardboard having dimensions 54 cmby 54 cm (w = 54) by cutting out squares of area x2 from each corner, as shown in thefigure below.

(a) Express the volume V (in cubic centimeters) of the boxas a function of x.

(b) State the domain of V

3 Linear Functions

1. Answer the following questions concerning the line that is graphed below.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(a) Find the coordinates of the y-intercept.

(b) Find the coordinates of the x-intercept.

(c) Find the slope of the line.

(d) Write the equation of the line in slope-intercept form.

(e) Find the equation of the line parallel to the given line that passes through the origin,and graph the line on the same graph.

2. Answer the following questions concerning the line that is graphed below.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(a) Find the coordinates of the y-intercept.

(b) Find the coordinates of the x-intercept.

(c) Find the slope of the line.

(d) Write the equation of the line in slope-intercept form.

(e) Find the equation of the line perpendicular to the given line that passes through theorigin, and graph the line on the same graph.

3. Find the equation of the line through the points (0,−4) and (−2, 2). Write the equationin slope-intercept form.

4. Find the equation of the line through the point (−2, 4) that is perpendicular to the line2x+ 11y = 4. Write your answer in slope-intercept form.

5. (a) A line passing through the point (−1, 0) has slope m = 3. Find the equation of theline in slope intercept form.

(b) Find the slope and y-intercept of the line passing through the point (−6, 4) that isperpendicular to the line described in part (a).

(c) Graph both lines on the same graph.

6. Find the equation of the line through the point (−7,−3) that is parallel to the line9x+ 2y = 3. Write your answer in slope-intercept form.

7. The rate at which water evaporates from a certain reservoir depends on the air tempera-ture. The table below shows the number of acre-feet (af) of water per day that evaporatefrom the reservoir for various temperatures in degrees Fahrenheit.

Temperature, ◦F af40 57660 153670 201690 2976

(a) Find a linear model for E(T ), the number of acre-feet of water that evaporate as afunction of temperature, T .

(b) Explain the meaning of the slope of this line in the context of this problem.

(c) Assuming that water continues to evaporate at the same rate, how many acre-feet ofwater will evaporate per day when the temperature is 78◦F?

8. A magazine company had a profit of $37800 per year when it had 13000 subscribers.When it obtained 15000 subscribers, it had a profit of $49800. Assume the profit P is alinear function of the number of subscribers s.

(a) Find the function P .

(b) What will the profit be if the company obtains 27000 subscribers?

(c) What is the number of subscribers to break even? (Round to the next highestsubscriber number if the number is not whole).

9. Julie opened a lemonade stand and found that daily her profit is a linear function of thenumber of cups of lemonade sold. When she sells 220 cups of lemonade, she makes $30and when she sells 320 cups of lemonade, she makes $60.

(a) Find the profit function.

(b) How many cups of lemonade does Julie need to sell to break even on a given day?

(c) How many cups of lemonade does Julie need to sell to make $240 in a day?

(d) How much would Julie make on a day when she sells 1500 cups of lemonade?

4 Quadratic Functions

1. A quadratic function f is graphed below. Use the graph to answer the following questions.

(a) Given that you know graph is a translation of either y = x2 or y = −x2. Which isit? Explain.

(b) Find the vertex of the quadratic function f .

(c) What is the axis of symmetry of the graph of f?

(d) Does f(x) have a maximum value? If so, at what value of x does it occur, and whatis the maximum value?

(e) Does f(x) have a minimum value? If so, at what value of x does it occur, and whatis the maximum value?

(f) Find the range of f(x). Express your answer in interval notation.

(g) Find f(x), that is, write the expression quadratic function f .

−10−8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

2. A quadratic function f is graphed below. It is a translation of the graph of y = −2x2.Use the graph to answer the following questions.

(a) Find the vertex of the quadratic function f .

(b) What is the axis of symmetry of the graph of f?

(c) Does f(x) have a maximum value? If so, at what value of x does it occur, and whatis the maximum value?

(d) Does f(x) have a minimum value? If so, at what value of x does it occur, and whatis the maximum value?

(e) Find the range of f(x). Express your answer in interval notation.

(f) Find f(x), that is, write the expression quadratic function f .

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

3. For this problem, f is the quadratic function

f(x) = −3x+ 6x+ 2

(a) Find the vertex of the graph of f .

(b) Write the quadratic function f(x) in standard form.

(c) Does the graph of f open upward or downward?

(d) Does f have a maximum value? If so, what is it?

(e) Does f have a minimum value? If so, what is it?

(f) Find the range of f . Write your answer in interval notation.

4. Consider the quadratic function f(x) = −(x− 4)2 + 4.

(a) Find the vertex of f(x).

(b) Does the graph of f(x) open upward or downward.

(c) Does f(x) have a maximum value? If so, what is it?

(d) Does f(x) have a minimum value? If so, what is it?

(e) Find the range of f(x). Express your answer in interval notation.

(f) Sketch the graph of f(x).

5. Find the coordinates of the vertex of the graph of the quadratic function f defined by

f(x) = 3x2 + 3x+ 1

6. (a) Find the coordinates of the vertex of the graph of the quadratic function f definedby

f(x) = −3x2 + 8x+ 9

(b) Find the range of the quadratic function f from (a). Express your answer in intervalnotation.

(c) Write the quadratic function f in standard form.

7. The height above the ground, in feet, of a projectile launched with an initial velocity of128 feet per second from an initial height of 12 feet above the ground is a function oftime t in seconds, given by

h(t) = −16t2 + 128t+ 12.

(a) Find the time t when the projectile reaches its maximum height.

(b) Find the maximum height attained by the projectile.

(c) Find the time t when the projectile hits the ground (has a height of 0 feet). Expressanswer in seconds to the nearest decimal place.

8. Find two numbers whose difference is 40 and the sum of whose squares is a minimum.

9. A farmer has 852 feet of fencing with which to make a rectangular enclosure that will besubdivided into two separate enclosures (see the figure below).

(a) Write the length l as a function of width w.

(b) Write the total area as a quadratic function of w.

(c) Find the dimensions of the enclosure that will produce the greatest enclosed area.

5 Properties of Graphs

1. Determine whether the following functions are even, odd, or neither.

(a) f(x) = 4x7 + 8x− 5

(b) g(x) = 7x4 − 8x2 − |x|+ 3

(c) h(x) = 8x7 − 3x3 + x

2. The graph of two different functions are given below in (a) and (b).

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) Is the function graphed above in (a) even, odd, or neither? Explain your answer.

(ii) Is the function graphed above in (b) even, odd, or neither? Explain your answer.

3. Which of the symmetries (over x-axis, y-axis, origin) are possessed by the graphs of thefollowing equations? Explain your answers.

(a) x8 + |y| = y5.

(b) 8|x|3 − 5|y|3 = −7.

(c) y3 = 8x+ 5.

(d) 8y + 5x = 0.

4. The graph of f(x) = |x| is given below. Find the function g(x) whose graph is the graphof f shifted horizontally 2 units to the left and vertically 3 units up, and then graph gon the graph below.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

f(x)x

y

5. The graph of f(x) = x3 is given below. Find the function g(x) whose graph is the graphof f shifted horizontally 2 units to the left and vertically 2 units up, and then graph gon the graph below.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

f(x)

x

y

6. The graph of f(x) = |x| and the graph of another function g(x) are given below on thesame axes.

(a) Describe how the graph of g(x) relates to the graph of f(x) in terms of horizontaland vertical translations.

(b) Find the function g(x): g(x) =

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

f(x)g(x)

x

y

7. The graph of a function f(x) is given below. Describe how the graph of the functiong(x) = f(x+ 1) + 1 relates to the graph of f , and then sketch the graph of g(x).

−6 −4 −2 2 4 6

−8

−6

−4

−2

2

4

6

8

x

y

8. The graph of the function f given below was obtained from horizontal and verticaltranslations of the graph of y =

√x. Find the function f(x).

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

For your reference, the points (−2,−2), (−1,−1), (2, 0), and (7, 1) were plotted on thegraph of f .

9. The graph of y = f(x) is given below in (a) and (b).

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

On the graph of (a) above, sketch y = f(x − 4), and on the graph (b) above, sketchy = f(x− 4) + 5.


(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

On the graph of (a) above, sketch y = −f(x), and on the graph (b) above, sketchy = f(−x).

11. The graph of y = f(x) is given below in (a) and (b). Assume the function f is periodic,that is, the shape of its graph keeps repeating.

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

On the graph of (a) above, sketch the graph of 12f(x). On the graph of (b) above, sketch

the graph of f(12x).

12. The graph of the function f given below was obtained by reflecting the graph of y =√x

over the x-axis, and performing horizontal and vertical translations on the reflected graph.Find the function f(x).

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

For your reference, the points (−5, 6), (−4, 5), (−1, 4), and (4, 3) were plotted on thegraph of f .

13. The graph of a function f(x) is given below. Describe how the graph of the functiong(x) = −f(x+ 2) relates to the graph of f , and then sketch the graph of g(x).

−6 −4 −2 2 4 6

−8

−6

−4

−2

2

4

6

8

x

y


(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) On the graph of (a) above, sketch the graph of y = 3f(x).

(ii) On the graph of (b) above, sketch the graph of y = 3f(x) + 1.

15. The graph of g in (a) was obtained by reflecting the graph of y =√x over the y-axis.

The graph of h in (b) was obtained by horizontal and vertical translations of the graphof g from (a). Find the functions h and g. (The points (0, 0) and (5,−5) are indicatedon the graphs of g and h respectively for your reference.)

(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

g

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

hx

y

16. The graph of y = |x| along with another graph is given in each of the graphs below in(a) and (b). In each case, the other graph is a result of reflections and translations ofy = |x|.(a) (b)

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

(i) Find the equation of the other graph in (a).

(ii) Find the equation of the other graph in (b).

6 Algebra of Functions

1. Let f(x) =√

2x+ 9 and g(x) =√

3− x. Find the domains of f + g, f − g, fg andf

g.

Write your answers in interval notation.

2. Let f(x) =√

64x+ 5 and g(x) = 4x− 1. Find f + g, f − g, fg andf

g.

3. Let f(x) = 3x2 − 4x+ 3 and g(x) = 4x− 9.

(a) Find (f ◦ g)(x)

(b) Find (g ◦ f)(x)

4. Let f(x) = 17x7 − 7 and g(x) =

(x+ 7

17

) 17

.

(a) Find (f ◦ g)(x).

(b) Find (g ◦ f)(x).

5. Let f(x) = 5x2 − 4 and g(x) = |x− 2|, find

(a) (g ◦ f)(−3) (b) (f ◦ g)(−3) (c) (f ◦ g)(0) (d) (fg)(1) (e) (f + g)(1)

6. Let f(x) = 6x2 + 6x− 5. Find and simplify the difference quotientf(x+ h)− f(x)

h.

7. Let f(x) = 4x2 − 7x. Find and simplify the difference quotientf(x+ h)− f(x)

h.

8. A water tank has the shape of a right circular cone with height 18 feet and radius 6 feet.Water is running into the tank so that the radius r (in feet) of the surface of the wateris given by r = 0.4t, where t is the time (in minutes) that the water has been running.See the diagram below which is not to scale, where A = 6 and B = 18.

(a) The area of the surface of the water is A = πr2. Find A(t) and use it to determinethe area of the surface of the water when t = 4 minutes. Express answer in square feet,rounded to two decimal places.

(b) The volume of the water is given by V = 13πr2h. Find V (t) and use it to determine

the volume of the water when t = 5 minutes. Express answer in cubic feet, rounded totwo decimal places.

9. Write the definition of the piecewise defined function f whose graph is given below.

−6 −4 −2 2 4 6

−6

−4

−2

2

4

6

x

y

math 121. practice problems from chapter 2 fall...

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