Name:____________________ Math 4 January 29/30, 2015 Review for Cumulative Test #2 Review problems for February 4 cumulative test Part I. Functions:

1. Describe a sequence of transformations to turn the graph of f x( ) into the graph of g x( ) .

a. f x( ) = 2x2 −1, g x( ) = −2 x + 7( )2

b.

€

f (x) = 2x , g(x) = 3⋅2x+4 +1

2. Let ( ) ( ) xxgxxf =+= and 52 . Find ( )( )xfg ! and ( )( )xgf ! .

3. Suppose g x( ) = 2x +3x − 5

.

a. Find g−1 x( )

b. Find the range of g x( ) . c. Find limx→∞

g x( ) .

Part II. Polynomials

4. A graph of the function f x( ) = x4 + 6x3 +10x2 + 6x + 9 is shown to the right.

a. Factor f(x) as much as possible in the real number system.

b. Factor f(x) as much as possible in the complex number system.

5. Let f x( ) = −x5 + 2x2 −3

a. Without using a calculator, use limit statements to describe the left- and right-end behavior of f(x).

b. What is the remainder when f(x) is divided by (x + 2)? Is (x + 2) a factor of f(x)? c. Find the real zeros of f(x). You may use your calculator.

 Part  III.  Rational  Functions

6.          Consider the function: f x( ) = x −3x2 + x −12

a. What is the domain of f(x)? b. Does f(x) have any holes? If so, find the x and y coordinates of the hole(s). c. Does f(x) have vertical asymptotes? If so, write the equation(s) of the asymptote(s). d. Describe the end behavior of f(x) using limit statements. e. Use limit statements to describe the behavior of f(x) at the discontinuities you identified in

parts b. and c.

 

Name:____________________ Math 4 January 29/30, 2015 Review for Cumulative Test #2

7. Consider a rational function, f(x), with the following properties: Vertical asymptotes at: x = 3 and x = − 4 limx→∞

f x( ) = 2 and limx→−∞

f x( ) = 2

f(−1) = 0 and f(−6) = 0. a. Find a possible function formula for f(x). b. Sketch a graph of f(x).

Part IV. Exponential and Logarithmic Functions 8. Suppose f x( ) = a ⋅bx where f (0) = 3 and f (3) = 3/64.

a. Find a function formula for f(x). b. If x is time measured in minutes, by what percentage (%) does f decrease each minute?

9. Strontium-90 is a radioactive substance that decays exponentially according to the equation

A x( ) = A0e−.0244x , where x is the time in years, A(x) is the amount present, and A0 is the amount present initially (time = 0). To answer these questions, assume that when time = 0, there are 400 grams of Sr-90 present.

a. How much Sr-90 will be left after 12 years? b. When will 100 grams be left? (Solve algebraically.) c. What is the half-life of Sr-90? d. Find A−1 x( ) .

e. What is A−1 50( ) ? Explain the meaning of your answer in the context of the problem.

10. Without using your calculator, find the exact values:

a. 52log5 6 b. log5 45− 2 log5 3

Part V. Trigonometry 11. The point (−1, 1) is on the terminal side of an angle, θ, in standard position.

a. Give the smallest positive angle measure for θ in both degrees and radians. b. Find the six trig functions of θ

12. Evaluate the following without using your calculator.

a. cot 3π( ) . b. sin 11π6

!

"#

$

%& c. sec −

2π3

"

#$

%

&'

13. Directions: Find tan

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θ and csc

€

θ given each set of information below:

a.

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cosθ =57

, tanθ < 0

b.

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cotθ is undefined and

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cosθ > 0  

Name:____________________ Math 4 January 29/30, 2015 Review for Cumulative Test #2 ANSWERS: Part I: 1. Describe a sequence of graphical transformations to turn the graph of ( )f x into the graph of ( )g x .

a. The left shift can be done in any order, but the order matters in the vertical direction.

1. Shift left by 7 units. 2. Shift up 1. 3. Reflect over the x axis. b. 1. Left by 4. 2. Stretch vertically by a factor of 3 3. Up by 1

2. ( )( ) 5)5())(( 22 +=+== xxgxfgxfg o ( )( ) 55)()())(( 2 +=+=== xxxfxgfxgf o 3. a. g−1 x( ) = 5x+3

x−2

b. Since the range of g(x) is the same as the domain of g−1 x( ) , the range of g(x) is: −∞, 2( )∪ 2,∞( )

c. One method: g(x) has a horizontal asymptote at y = 2, so limx→∞

g x( )= 2.

Another method: g-1(x) has a vertical asymptote at x = 2, so g–1(x) has a horiz. asymptote at y = 2. Part II:

4. f x( ) = x4 + 6x3 +10x2 + 6x + 9 . From the graph, we can see that x+3 is a factor with multiplicity = 2. Using long division:

a. f (x) = (x +3)2(x2 +1)

b. f (x) = (x +3)2(x + i)(x − i)

5. f x( ) = −x5 + 2x2 −3

a. End behavior is Up/Down: limx→−∞

f x( ) =∞ and limx→∞

f x( ) = −∞

b. The remainder is: f (−2) = 37. Therefore, x+2 is not a factor.

c. One real zero: x = −1. Part III:

6. f x( ) = x −3x2 + x −12

a. Domain: (−∞,−4)∪ (−4,3)∪ (3,∞)

b. Hole at: 3, 17

!

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$

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c. Vertical asymptote at x = −4.

d. limx→∞

f x( ) = 0 and limx→−∞

f (x) = 0

e. limx→−4−

f (x) = −∞ , limx→−4+

f (x) =∞ , limx→3

f (x) = 17

 

Name:____________________ Math 4 January 29/30, 2015 Review for Cumulative Test #2

7.

a. f (x) = 2(x +1)(x + 6)(x −3)(x + 4)

b. Check your graph with your calculator. Part IV:

8.

a. f (x) = 3 14!

"#$

%&x

b. f (x) = 3 14!

"#$

%&x

= 3 .25( )x , so decreases by 75% each minute.

9. A x( ) = 400e−.0244x

a. A(12) = 298.5 grams

b. x = ln(1 / 4)−.0244

= 56.82 years

c. 28.41 years.

d. A−1(x) = −1.0244

ln x400"

#$

%

&'

e. A−1(50) = 85.22 . In 85.22 years, 50 of the original 400 grams of Sr-90 will be left.

10. a. 36 b. 1 PART V:

11. The point is: (−1,1) a. θ = 135o or 3π/4 radians

b. sinθ = yr=12

cscθ = ry= 2

cosθ = xr=−12

secθ = rx= − 2

tanθ = yx= −1 cotθ = x

y= −1

12. a. Undefined b. − 12

c. −2

13. a. tanθ = − 245

, cscθ = 7− 24

b. tanθ = 0, cscθ is undefined