# College Algebra Chapters 3 & 4 Homework

**Show your work or give some kind of justification for each answer!**

1. Complete the square to find the vertex of the

parabola

f(x) = 3x^{2} - 5x + 1

f(x) = _______________________(6)

Vertex___________

2. Use synthetic division to show that 3 is a

zero of the function

f(x) = 2x^{3} - 5x^{2} - 7x +12

(Write a sentence to explain HOW YOU

KNOW that 3 is a zero.) Then use that

information to find exact expressions for all

of the zeros of the function.

________________(7)

3. f(x) = - x^{5} + 2x + 1

g(x) = 7x^{4} + 1

h(x) = - x^{2} - 4

G(x) = x^{7} + x4 + 1

H(x) = 4x^{3} - x^{2} + 5x

From the list of functions above, identify the

function or functions whose graphs display

the given left-hand and right-hand behavior.

a) Up to the left and down to the right.

__________(3)

b) Down to the left and down to the right.

__________(3)

4. Identify the type of graph (line, circle ,

parabola, etc.) of each of the following:

a) 2x^{2} + 5y^{2} - x - 12 = 0

_______________(2)

b) 7x^{2} + 3x + y - 6 = 0

_______________(2)

5. Find all vertical, horizontal, and slant

asymptotes of these functions.

_______________________(6)

_______________________(6)

6. Find all of the rational numbers which ,

according to the Rational Zero Test, could

be rational zeros of the function

g(x) = 6x^{5} + x^{4} + 7x^{3} - 4x^{2} - 5x - 7

______________________________(5)

7. Refer to the function in the previous

problem. According to Descartes' Rule of

Signs , how many positive real zeros and

how many negative real zeros might the

function have?

positive__________(6)

negative__________

8. A candy manufacturer believes that the

demand (D) for a certain kind of candy will

be directly proportional to the amount (in

thousands of dollars) spent on advertising

(A) and inversely proportional to the square

of the price in dollars (p).

a) Write an equation that expresses this

relationship.

_____________(4)

b) If D = 160 units when A = 90 and p = 3,

find the constant of proportionality.

___________(2)

9. Divide (6x^{3} - 16x^{2} + 17x -10) by (3x - 2)

_________________________________(4)

10. For

a) As x → -5^{+}, f(x) → ____ (3)

b) As x → + ∞, f(x) → ____ (3)

11. Find an equation of the parabola with vertex

at the origin and focus at (-5, 0).

___________________(5)

12. Given: f(x) = 6x^{4} - 5x^{3} - 15x^{2} - 5x - 21

a) Show that 3 is a upper bound of the real

zeros of the function. (Be sure to include

a sentence explaining your conclusion.)

_______________(3)

b) A graph of the function indicates that

there is a zero between -2 and -1, and

another zero between 2 and 3. Use the

Rational Zero Test to determine the

possible rational zeros that lie between -

2 and -1 or between 2 and 3.

_______________(3)

c) Given that both of the zeros described in

(b) are rational, find these zeros and all

other complex zeros of the function.

_____________________(5)

13. For the hyperbola with the equation

sketch the graph, including the asymptotes,

identifying the vertices, foci, and endpoints

of the transverse and conjugate axes.

( Coordinates may be rounded to the

nearest 0.1)

____________________(6)

14. Find the center, foci, and endpoints of the

axes of the ellipse

3x^{2} + y^{2} + 12x - 2y + 4 = 0

(Coordinates may be rounded to the

nearest 0.1)

____________________(6)

**Answers:
**Vertex: (5/6, -13/12)

2. [Syn. div. produces a remainder of 0.]

3. a) f b) h

4. a) ellipse b) parabola

5. a) H: y = 3/2 V: x = 0, x= 5

b) S: y= 3x - 20 V: x = -7

6. ±1, ±7, ±½, ±7/2, ±1/3, ±7/3, ±1/6, ±7/6

7. Pos: 1 Neg: 4 or 2 or 0

10. a) -∞ b) 2 —

11. y

^{2}= -20x

12.

a) [Do syn. div. All numbers in bottom row

are positive .]

b) -3/2 and -7/6; 7/3

13. Center: (3, -2)

a = 3, b = 4.5, c = 5.4

Vertices: (0, -2), (6, -2)

Ends of conjugate axes: (3, 2.5), (3, -6.5)

Foci: (-2.4, -2), (8.4, -2)

14. Center: (-2, 1)

a = 3, b = 1.7, c = 2.4

Vertices (major axis): (-2, -2), (-2, 4)

Ends of minor axis: (-3.7, 1), (-0.3, 1)

Foci: (-2, -1.4), (-2, 3.4)

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