Math reviwe questions)
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exam_4_review.pdf
Math 121 Exam 4 Review Problems
1. Sketch the graph of each function. Include the vertex, the y-intercept, and any x-intercepts. Show work.
2
( ) 2( 3) 1f x x= − + − ( ) ( 1)( 3)f x x x= − + 2
( ) 2 8 8f x x x= + +
2. Write the equation of each function.
3. Write the function in the general form, 2
,ax bx c+ + and find the vertex.
( ) 2( 7)( 3)f x x x= − +
4. Write the function in the general form, 2
,ax bx c+ + and then find the x-intercepts.
2
( ) 2( 3) 15f x x= − −
5. How many x-intercepts will each of the following functions have? Can you find the x-intercepts by factoring?
a. 2
( ) 3 6 8f x x x= + − b. 2
( ) 3 24 48f x x x= + + c. 2
( ) 3 9 8f x x x= + +
6. Given the quadratic equation:, 2
( ) 2 4 4h x x x= − + +
a. Complete the square and put the function in vertex form.
b. Find the x- intercepts using the quadratic formula
c. Graph the function, labeling the vertex, the x- intercepts and the y-intercept.
d. DOMAIN: ________________ RANGE: ___________________
7. Given the quadratic equation:, 21 3
( ) 2 2
g x x x= + −
a. Complete the square and put the function in vertex form.
b. Find the x- intercepts using the quadratic formula.
c. Graph the function, labeling the vertex , the x- intercepts and the y-intercepts.
d. DOMAIN: ________________ RANGE: ___________________
8. Write each formula as it is described.
a. The pressure, P, is directly proportional to the square of the temperature, T.
b. The pressure, P, is inversely proportional to the square root of the temperature, T.
9. Describe each formula using the language of proportionality
a. 2
12V rπ= where V is volume and r is radius of a cylinder.
b. 130V
P = where V is volume and P is pressure.
10. Sketch the graph of each function by finding x-intercepts, the y-intercept, and end behavior.
( ) ( 2)( 1)( 2)f x x x x= − − + 2
( ) .5( 2)( 1) ( 3)f x x x x= − − − + 3 2
( ) 6f x x x x= − + −
11. The intensity of light from a point source is inversely proportional to the square of the distance from the light
source.
a. If the intensity is 4 watts per square meter at a distance of 6 meters from the source, find the constant of proportionality, k, and write the formula for intensity, I, as a function of distance, d, from the source.
b. Find the intensity at a distance of 8 meters from the source.
c. If the distance from the source doubles, what is the effect on the intensity of the light? Answer in a clear sentence and show complete and persuasive mathematical work.
d. Find the inverse formula, d(I), and compose the two functions to demonstrate that they are truly inverse functions.
12. If the American Automobile company sells x cars, its profit is given by the function: 2
( ) 50 14000 20000P x x x= − + − dollars. Use the information you obtain below to choose scales for each axis and
sketch the graph of the profit function.
a. Find the number of cars that the company needs to sell in
order for their profit to be maximized.
b. What is that maximum profit?
c. Find the horizontal intercepts of the profit function.
d. Interpret your result for part c.
13. The period of a pendulum (the time elapsed during one complete swing) is directly proportional to the
square root of the length of the pendulum.
a. Express this proportionality with an equation.
b. If the pendulum length is doubled, what is the effect on the length of the period? Answer in a clear sentence and show complete and persuasive mathematical work.
c. Suppose a pendulum that is 3 feet long has a period of 0.5 seconds. Write the formula for period, P, as a function of length, L, of the pendulum.
d. Find the inverse formula, L(P), and compose the two functions to demonstrate that they are truly inverse functions.
14. Write each rational expression in factored form. Reduce the expression if possible.
a. 2
3
15
3 9
x
x x− b.
2
2
6 8
2 4
x x
x x
+ +
−
c. 2
2
6 8
4 5
x x
x x
+ +
− −
14. Sketch the graph of each function by finding x-intercepts, the y-intercept, and horizontal and vertical
asymptotes. Explain the relationship between the first two graphs and the function, 1( )g x x
= and write each
in ( ) cf x b x a
= + −
form. Use your graphing calculator to finalize your graph of the third function.
2 4
( ) 1
x f x
x − −
= +
3 1
( ) 2
x f x
x +
= −
2
( 3)( 1) ( )
( 2)
x x f x
x
− + =
+
17. Write each rational function in compound form and find the horizontal asymptote.
a. 2
2
5 2 7( ) 2 4 3
x xf x x x
− + =
− +
b. 2
2 7( ) 4 3
xf x x x
+ =
− +
18. The game commission introduces 50 deer into newly acquired game lands. The population N of the herd is modeled
by 10(5 2 )
1 .05
t N
t
+ =
+ where t is the number of years since the deer were first introduced.
a. How large will the population of deer be after 15 years?
b. What is the horizontal asymptote? ______________ What does the horizontal asymptote tell you about the deer
population?
19. A cannonball that is fired out to sea from a shore battery follows a parabolic trajectory given by the graph of the
equation 2
( ) 12 .01h x x x= − where h(x) is the height of the cannonball above the water when it has traveled a
horizontal distance of x feet.
a. What is the maximum height that the cannonball reaches?
b. How far does the cannonball travel horizontally before splashing into the water?
