review questions for math 102 ( college Algebra )

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MATH 102 Springs 2015 REVIEW FOR FIRST EXAM

Difference Quotient

1. For 2( ) 3 9 1f x x x= − + , find ( ) ( )f x h f x

h + −

and simplify completely.

2. For 8( )f x x

= , find ( ) ( )f x h f x

h + −

and simplify completely.

3. For ( ) 4f x x= , find ( ) ( )f x h f x

h + −

and simplify completely.

4. For 23( ) 7f x x= − , find ( ) ( )f x h f x

h + −

and simplify completely.

Domain of a Function

5. Find the domain of each of the following functions. Express your answers using interval notation.

a. 2

3 ( )

16 x

f x x −

= −

b. ( ) 2 9g x x= − c. 21 12 3( ) 5 4h x x x= − +

6. Find the domain of each of the following functions. Express your answers using interval notation.

a. 3

( ) 5

x f x

x −

= −

b. ( ) 2 8h x x= − c. 2( ) 5 6 x

g x x x

= − +

7. Find the domain of each of the following functions. Express your answers using interval notation.

a. 2( ) 1h x x x= − − − b. 3

( ) 5 10 x

f x x −

= −

c. 2 4

( ) x

h x x −

8. Find the domain of each of the following functions. Express your answers using interval notation.

a. 2

1 ( )

4 x

h x x

− =

− b.

10 ( )

2 1 f x

x =

+ c. ( ) 5h x =

Piecewise Functions

9. Let

1 2

0 ( ) 1 0

1 0

x if x f x if x

x if x

⎧ > ⎪

= =⎨ ⎪− − <⎩

a. Find (6)f . b. Find the y-intercept. Write your answer as an ordered pair. c. Find the x-intercept. Write your answer as an ordered pair. d. Sketch the graph of ( )f x . Label x and y intercepts.

10. Let 2 3 1

( ) 1

x if x f x

x if x

− + ≤⎧⎪ = ⎨

>⎪⎩

a. Find (9)f b. Sketch the graph of ( )f x . Label with ordered pairs the points on the graph where

1, 0, and1x = − .

11. Let 2 6 2 1

( ) 7 1 2 5 1

x if x f x if x

x if x

+ − ≤ <⎧ ⎪

= =⎨ ⎪− + >⎩

a. Find (3)f . b. (3 pts.) Find the y-intercept. Write your answer as an ordered pair. c. (3 pts.) Find the x-intercept. Write your answer as an ordered pair. d. (7 pts.) Sketch the graph of ( )f x . Label x and y intercepts.

Information from a graph

12. Use the graph of the function f below to answer the following:

a. What is ( 1)f − ? b. For how many values of x is ( ) 2f x = ? c. For which value of x does f have a local maximum? d. What is the average rate of change of f on the interval [ ]4, 2− − ? e. For which values of x is ( ) 0f x ≥ ?

13. Use the graph of the function f below to answer the following:

a. What is (4)f ? b. What is the average rate of change of f on the interval [ ]1,2− ? c. What is a local maximum value of f? d. On what interval is f increasing? e. What is the domain of f? f. For which values of x is ( ) 0f x ≤ ?

1 2 3

3 2

14. Use the graph of the function f below to answer the following:

a. What is (3)f ? b. What is the average rate of change of f on the interval [ ]5, 1− − ? c. What is a local minimum value of f? d. On what interval is f decreasing? e. What is the domain of f? f. For which values of x is ( ) 2f x > ?

Graphing Using Transformations

15. Sketch the graph of 12( ) 1f x x= − + using transformations. Start by sketching the corresponding basic function and then show each stage of the transformation (including each equation). Label at least 2 points on the final graph. 16. Sketch the graph of 3( ) ( 3) 2f x x= − − + using transformations. Start by sketching the corresponding basic function and then show each stage of the transformation (including each equation). Label at least 2 points on the final graph. 17. Sketch the graph of ( ) 4 3f x x= − − using transformations. Start by sketching the corresponding basic function and then show each stage of the transformation (including each equation). Label at least 2 points on the final graph. Modeling 18. A wire of length 5x is bent into the shape of a square. Express the area A of the square as a function of x.

19. Let ( , )P x y= be a point on the graph of y x= . Express the distance d from P to the point (3,0) as a function of x. Simplify your answer if possible.

1 2 3

20. Two bicycles leave an intersection at the same time. One is headed due north at a constant speed of 10 miles per hour and the other is headed due east at a constant speed of 15 miles per hour. Express the distance d (in miles) between the bicycles as a function of time t (in hours). (Let 0t = be the moment the cars leave the intersection.) Simplify your answer.

Average Rate of Change

21. Find the average rate of change of the function 3( ) 3 2f x x= − on the interval [ ]4,15 . Simplify your

answer completely.

22. Find the average rate of change of the function ( ) 4 2f x x= − on the interval 326,−⎡ ⎤⎣ ⎦ . Simplify your answer completely.

23. Find the average rate of change of the function ( ) 5 2f x x= − on the interval [ ]122,− . Simplify your answer completely. 24. Find the average rate of change of the function 2( ) 3 2f x x x= − − + on the interval [ ]2,4− . Simplify your answer completely.

Using functions as models to make predictions and draw conclusions 25. The function ( ) .01 58f x x= + models the global average temperature of the Earth in degrees

Fahrenheit, x years after January 1, 1995. a. Is this function increasing, decreasing, or constant? b. According to this model, in what year will the global average temperature of the Earth reach 59 degrees Fahrenheit? c. Assuming this model is accurate, what is its implied domain? d. What is the average rate of change of this function from July 2, 1999 to August 14, 2007?

26. When the San Diego Chargers punter, Mike Scifres, kicks the ball, its flight follows a path given by 21

2( ) 20H x x x= − + where H is the height of the ball in feet above the ground and x is the horizontal distance of the ball in yards from its starting point. Answer each of the following and include the correct units with each answer.

a. What is the height of the ball after it has traveled a horizontal distance of 10 yards? b. How far away from the starting point will the ball be when it hits the ground? c. What is the maximum height attained by the ball? d. What is the practical domain of this function?

27. The function 12( ) 24f x x= − + models the percentage of U.S. adults who smoked cigarettes, x years after January 1, 1997.

a. Is this function increasing, decreasing, or constant? b. According to this model, in what year will zero percent of U.S. adults smoke cigarettes? c. Assuming this model is accurate, what is its implied domain? d. What is the average rate of change of this function from July 2, 1999 to August 14, 2007? 28. If an arrow is shot vertically from a bow with an initial speed of 192 ft./sec, the height of the arrow

can be modeled by the function 2( ) 16 224h t t t= − + , where ( )h t represents the height of the arrow after t seconds (assume the arrow was shot from ground level).

a. What is the arrow's height after 1 second? b. How long does it take the arrow to hit the ground? c. After how many seconds does the arrow attain its maximum height? d. What is the practical domain of h?

Identifying Properties of Quadratic Functions - For each of the following quadratic functions: a. Express in the form 2( ) ( )f x a x h k= − + by completing the square. b. On what intervals is f increasing and decreasing? c. Identify the maximum or minimum value of f. d. Write the equation of the axis of symmetry. e. Sketch the graph of f . Label the vertex, y-intercept, and axis of symmetry. f. Find the domain and range of f. 29. 2( ) 3 12 13f x x x= − + 30. 2( ) 2 16 33f x x x= + +

31. 2( ) 2 12 15f x x x= − + − 32. 2( ) 4 40 97f x x x= − + − Identifying Properties of Linear Functions - For each of the following quadratic functions: a. Identify the slope and y-intercept of the graph of f. b. Use the slope and y-intercept to sketch the graph of f. c. Is f increasing, decreasing, or constant? d. Find the average rate of change of f on the interval 313 ,1.34567⎡ ⎤⎣ ⎦. f. Find the domain and range of f.

33. 3

( ) 5 4

f x x= − + 34. ( ) 3f x = −

35. ( ) 3 4f x x= − 36. 2

( ) 4 5

f x x= −

Constructing and Using Linear and Quadratic Functions as Models 37. A company buys a new computer for $2000 on October 1st, 2012. Assume that the computer loses

value linearly and that in 4 years its value is $400. a. Express the value V (in dollars) of the computer as a function of time t (in years) . b. Use your answer from part (a) to find the value of the computer on January 1st, 2013.

38. A rectangular field is to be enclosed with 280 yards of fencing. One side of the field faces a river and does not need fencing.

a. Express the area of the field A as a function of its width x. b. For what value of x will the area be the greatest?

39. The monthly revenue generated by the sale of x wristwatches is given by 2( ) 90 .2R x x x= − . The

monthly cost of producing and selling x wristwatches is given by ( ) 28 1900C x x= + . What is the maximum profit that can be generated? (Hint: Profit is given by ( ) ( ) ( )P x R x C x= − ).