Math 13 questions step by step needed

Math 121G – Practice Exam 4

Quadratic Formula:

2 4

2 b b ac

x a

    Standard or Vertex form:

2 ( ) ( )f x a x h k  

1. [8 points] Sketch the graph of the function, ( ) .25( 6)( 2)f x x x   . Include the vertex, the y-intercept,

and any x-intercepts. Show work.

2. [8 points] Write an equation for the function shown. Show work.

3. [10 points] Given the quadratic equation:, 2

( ) 3 12 3h x x x    , complete the square and put the function in

vertex form. Then write the ordered pair for the vertex.

4. [4 points] Write the function, 2

( ) 2( 3) 5f x x    , in the general form, 2

.ax bx c  Show work.

5. [4 points] Use the discriminant to determine how many x-intercepts the function, 2

( ) 2 6 4.5f x x x  

will have. Show work.

6. [10 points] A cannonball that is fired out to sea from a shore battery follows a parabolic trajectory given by the graph

of the equation 2

( ) 8 11 .02h x x x   where h(x) is the height in feet of the cannonball above the water when it has

traveled a horizontal distance of x feet.

a. Find the vertex of the trajectory and explain what the vertex represents in terms of the cannonball’s flight.

b. Find the x-intercepts of the trajectory and explain what each x-intercept represents in terms of the cannonball’s

flight.

7. [6 points] Describe the formula using the language of proportionality.

31

27 V s where V is volume, in cubic yards and s is the length of the side of a cube, in feet.

8. [10 points] Write the graph of the function, 4 3 2

( ) 2 15f x x x x   in factored form. Then sketch the graph

using x-intercepts, the y-intercept, and end behavior. Verify your result using your graphing calculator.

9. The length of a pendulum is directly proportional to the square of the period (length of time of one swing).

a. [4 points] Express this proportionality with an equation.

b. [4 points] Suppose a pendulum that is 5 feet long has a period of 1 second. Write the formula for length, L, as a function of the period, P, of the pendulum.

c. [10 points] Find the inverse formula, P(L), and compose the two functions to demonstrate that they are truly inverse functions.

10. [4 points] Write the rational expression in factored form. Reduce the expression if possible. Show work.

2

2

7 12

4

x x

x x

 



11. [4 points] Write the rational function in compound form and find the horizontal asymptote.

2

5 3( ) 5 3

xf x x

 

.

12. [12 points] Sketch the graph of the function, 3 3

( ) 2

x f x

x 

 

, by finding x-intercepts, the y-intercept, and

horizontal and vertical asymptotes. Give the domain and range. Show work.

13. Psychologists have developed a mathematical model to predict memory performance as a function of the

number of trials, n, of a certain task. Consider the learning curve 0.6 0.95( 1)

1 0.95( 1)

n P

n

  

  where P is the

fraction of correct responses after n trials.

a. [6 points] What is the horizontal asymptote for this equation? ______________ What does it tell you

about the performance of the task?

b. [4 points] What is the vertical asymptote for this equation? ____________________ What does it tell you about

the performance of the task?