As $$x \to \infty$$, $$f(x) \to \infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Question 1

What is the vertex of the parabola given by $$f(x) = -7x^2$ + 3$$?

Accepted Answer

$$(0, 3)$$

Question 2

Find the axis of symmetry for the parabola $$f(x) = 3x^2$ + 3.$$

Accepted Answer

$$x = 0$$

Question 3

Determine the domain and range of the function $$f(x) = x^2$ - 2.$$

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$[-2, \infty)$$

Question 4

Which of the following polynomial functions has a graph symmetric with respect to the y-axis?

Accepted Answer

$$y = 4x^2$ + 2$$

Question 5

Determine if the function $$f(x) = x^4$ - 2x^2$ - 3 is $$even, odd, or neither.

Accepted Answer

Even

Question 6

What is the leading term, leading coefficient, and degree of the polynomial $$f(x) = 7x^8$ - 10 + 0.12x - x^3$?

Accepted Answer

Leading term: $$7x^8$$, Leading coefficient: $$7$$, Degree: $$8$$

Question 7

Use synthetic division to divide $$3x^2 - 3x + 6$ by x + 2$. What is the remainder?

Accepted Answer

$$6$$

Question 8

Given $$f(x) = 2x^3$ + 5x^2$ + 12x - 8$$, which of the following is NOT a possible rational zero according to the Rational Root Theorem?

Accepted Answer

$$\frac{3}{2}$$

Question 9

Use Descartes' Rule of Signs to determine the possible number of positive real zeros for $$f(x) = 8x^3$ - 6x^2$ + 8x + 8.$$

Accepted Answer

2 or 0

Question 10

Find all rational zeros and factor $$f(x) = x^3$ + 2x^2$ - 16x - 32.$$

Accepted Answer

Rational zeros: $$-2, -4, 4$$; Factored form: $$(x + 2)(x + 4)(x - 4)$$

Question 11

State the domain of the rational function $$f(x) = \frac{7x - 6}{8x - 16}.$$

Accepted Answer

$$x 
eq 2$$

Question 12

Find the vertical asymptote(s) of $$f(x) = \frac{x + 9}{x^2$ - 9}.$$

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 13

Find the horizontal asymptote of $$f(x) = \frac{3x^2$ - 4x - 6}{5x^2$ - 5x + 5}.$$

Accepted Answer

$$y = \frac{3}{5}$$

Question 14

Find the slant (oblique) asymptote of $$f(x) = \frac{x^2$ + 4x + 3}{x - 9}.$$

Accepted Answer

$$y = x - 5$$

Question 15

Solve the inequality $$\frac{x - 3}{x^2$ - 1} \u003c 0$$ and describe the solution set.

Accepted Answer

$$(-\infty, -1) \cup (1, 3)$$

Question 16

Which of the following describes the end behavior of $$f(x) = 7x^3$ - 3x^2$ - 5x + 1$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 17

Given $$f(x) = 5(x - 6)^2$(x - 6)^3$, what is the zero and its multiplicity?

Accepted Answer

Zero: $$x = 6$$, Multiplicity: $$5$$

Question 18

Use the intermediate value theorem to show that $$f(x) = x^3$ + 2x^2$ - 16x - 32$$ has a real zero between $$x = 2$$ and $$x = 3.$$

Accepted Answer

$$f(2)  0$$, so there is a zero between $$x = 2$$ and $$x = 3$$

Question 19

Solve the inequality $$5x \u003c 8x$$ and graph the solution set.

Accepted Answer

$$x \u003e 0$$

Polynomial and Rational Functions: Key Concepts and Problem-Solving Strategies

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