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

Question 1

Given the polynomial $$f(x) = (x + 2)^3$ - 3$$, what is the parent function, and what transformation has been applied to it?

Accepted Answer

Parent function: $$f(x) = x^3$; transformation: shift left 2 units and down 3 units

Question 2

What is the degree of the polynomial $$f(x) = x^4$ - 2x^2$ + 1$$?

Accepted Answer

Degree 4

Question 3

A polynomial function has zeros at $$x = -2$$, $$x = 3$$, and $$x = 6. $$Which of the following is a possible function?

Accepted Answer

$$f(x) = (x + 2)(x - 3)(x - 6)$$

Question 4

Given $$f(x) = -2(x - 1)^2$ + 4$$, which transformation is applied to the parent function $$f(x) = x^2$?

Accepted Answer

Reflect over the x-axis, vertically stretch by 2, shift right 1 unit, and up 4 units

Question 5

Which of the following could be the graph of a polynomial with zeros at $$x = -1$$, $$x = 2$$, and $$x = 4$$?

Accepted Answer

A graph crossing the x-axis at $$x = -1$$, $$x = 2$$, and $$x = 4$$

Question 6

If $$f(x) = x^3$ - 2x^2$ + x - 5$$, what is the y-intercept of the function?

Accepted Answer

$$-5$$

Question 7

Which of the following is the domain of the rational function $$f(x) = \dfrac{2x + 1}{x^2$ - 4}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 8

Given $$f(x) = \dfrac{3x}{x - 2}$$, what is the vertical asymptote?

Accepted Answer

$$x = 2$$

Question 9

For the rational function $$f(x) = \dfrac{x^2$ - 9}{x + 3}$$, what is the x-intercept?

Accepted Answer

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

Question 10

Which of the following describes the end behavior of $$f(x) = -x^4$ + 2x^2$ - 1$$?

Accepted Answer

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

Question 11

A chemical reaction rate is modeled by $$R(t) = 5t^2$ - 3t + 2. $$What is the instantaneous rate at $$t = 1$$?

Accepted Answer

$$R(1) = 5(1)^2$ - 3(1) + 2 = 4$$

Question 12

Which of the following is the correct factorization of $$x^3$$ - $$4x^2 + 4x$?

Accepted Answer

$$x(x^2 - 4x + 4$$)$$

Question 13

A company’s profit is modeled by $$P(x) = -2x^2$ + 8x - 6. $$What is the maximum profit, and at what value of $$x$$ does it occur?

Accepted Answer

Maximum at $$x = 2$$, $$P(2) = 2$$

Question 14

Which of the following is the correct graph for $$f(x) = -x^2$ + 4$$?

Accepted Answer

A downward opening parabola with vertex at $$(0, 4)$$

Question 15

Given $$f(x) = x^3$ - 3x^2$ + 2x$$, which of the following is NOT a zero of the function?

Accepted Answer

$$x = -1$$

Question 16

For the rational function $$f(x) = \dfrac{x^2$ - 1}{x - 1}$$, what is the simplified form and its domain?

Accepted Answer

Simplified: $$f(x) = x + 1$$, domain: all real numbers except $$x = 1$$

Question 17

Which of the following is the correct equation for a polynomial with zeros at $$x = 1 ($$multiplicity 2) and $$x = -3$$?

Accepted Answer

$$f(x) = (x - 1)^2$(x + 3)$$

Question 18

A rational function $$f(x) = \dfrac{2x^2$ + 3x - 2}{x - 1}$$ has a hole at $$x = 1. $$What is the y-value of the hole?

Accepted Answer

$$f(1) = 2(1)^2$ + 3(1) - 2 = 3$$

Question 19

Which of the following is the correct parent function for $$f(x) = (x - 2)^5$ + 4$$?

Accepted Answer

$$f(x) = x^5$

Polynomial and Rational Functions: Study Notes for College Algebra

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