Math 121 Quiz 4 Extra Problems sections 3.2-4.2

As always you are responsible for all concepts covered in quizzes, activities, handouts and in the textbook. There may be problem types on the test that are not covered here.

1.) Match the equations to their graphs below. Choose the best answer. Assume the scale on each graph is the same.

(i.) _____           (ii.) _____

                    (iii.) _____       (iv.) _____

2.) Give the equation of the quadratic for each of the following.

  1. As pictured:
  2. vertex at (2, -1) and also goes through the point (0, 2)

3.) For what values of a does the quadratic function y = ax2-2x + 3 not cross the x-axis.

4.) Suppose you have a 6 ft. by 8 ft. built-in swimming pool. You want to put in a rectangular concrete border all the way around. You can afford 72 square feet worth of concrete. How wide is your border?

5.) Write the equation y = 3x2 - x + 2 in standard form and identify the vertex.

6.) Solve for all real solutions:

a.) b.) (x + 2)(x - 5) = -10

7.) Find all complex solutions to the following equations using any method you wish. Show all written work.

a.) b.)
c.) d.) 4x4 - 15x2 - 4 = 0

8.) Solve for x exactly: a.) 8x6 + 63x3 = 8     b.) 2x2 + 5x –1 = 0

9.) Write the equation y = 3x2 − x + 2 in the form y = a(x − h)2 + k and identify the vertex.

10.) An open box is made from a rectangular piece of cardboard that has dimensions 40 cm. by 60 cm. The box is made by removing square corners of area x2 and turning up the sides.

(a.) Find a function for the volume V(x) of the box in terms of x.

(b.) Restrict the domain of V(x) to possible values of x that are reasonable for this problem.

(c.) Find all possible values of x that will result in a box with volume greater than 5000 cm3.

11.) Use the Intermediate Value Theorem to show that f(x) = 5x4 + 2x3 - 3x2 + 2x - 4 has a zero between x = 0 and x = 1.

12.) Solve the following inequality and write your answer in interval notation: 5|3 - 2x| + 1 > 96

13.) Solve the following inequalities algebraically and sketch a graph of the solution. Write your answers in interval notation.

14.) Suppose Tenera has 15 feet of fencing to enclose a rectangular garden next to a building. She plans to use the fencing on three sides and the building on the fourth side of the rectangle. She also plans to divide the rectangle into two parts with the fencing as pictured.

  1. Write a function for the area of the enclosed region in terms of the length x of an adjacent side to the building.
  2. Find the value of x that will maximize the area.

15.) Solve for all real values of x:

  1. |2x + 1| + 2 = 3

16.) Find the third-degree polynomial function in the picture.

17.) Let the following equations represent polynomials where the leading and last terms are given, but the middle terms are unknown. Choose the best graph for each polynomial. Do not use any graph more than once.

f(x) = -6x6 + ... + 5 h(x) = 6x6 + ... -5 l(x) = 6x4 + ... + 5
g(x) = -6x7 + ... - 5 k(x) = -6x5 + ... + 5 m(x) = 6x3 + ... - 5

18.) Graph without a calculator: f(x) = -2x5 - 6x4 + 8x3 + 24x2

Answers:

1.) (i.) f    (ii.) a    (iii.) e    (iv.) h

2.)a.) y = -2(x – 1)2+ 2    b.) y = ¾(x – 2)2 - 1

3.) a > 1/3

4.) 2 feet                    5.)

6.a.) 0,     b.) 0, 3

7.) a.) x = 9 ;    b.) no solution     c.) x = ±8     d.) x = ±i/2, ±2

8.) a.) x = ½ or -2    b.)

9.)    vertex =
10.) a.) V(x) = x(40 - 2x)(60 - 2x)

    b.) 0 < x < 20

c.)
11.) f(0) = -4, f(1) = 2 so by the I.V. Thm. We have f(x) = 0 for some x between 0 and 1.
12.) (-∞, -8] ∪ [11, ∞)

13.)
  1.    
  2.   (-∞, 4/15] ∪ [4/3, ∞)  
  3.   [-1, 5)  
14.)
  1. A(x) = (15 –3x)x = 15x –3x2
  2. x coordinate of vertex is: -b/(2a) = -15/(2(-3)) = 5/2
    x = 2.5 feet
15.)
  1. 0, -1
  2. 1
  3. -3125
  4. no real solns
  5. 36
16.)

17.) f: e,    h: b,    l: d,
g: a,    k: c,    m: f

18.) f(x) = -2x2(x − 2)(x + 2)(x + 3)