Q: Which of the following is the correct end behavior for the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

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

Question 1

A piecewise function is defined as follows:

$$
\begin{cases}
 2x + 1 & 	ext{if } x \u003c 0 \
 x^2$ & 	ext{if } x \geq 0
\end{cases}
$$

What is the value of $$f(-3)$$?

Accepted Answer

$$f(-3) = -5$$

Question 2

Given the polynomial $$f(x) = x^3$ - 4x^2$ + 4x$$, which of the following statements about its zeros is correct?

Accepted Answer

The polynomial has two real zeros, one of which has multiplicity 2.

Question 3

A polynomial $$f(x)$$ has a zero at $$x = 2$$ with multiplicity 3. How does the graph behave at $$x = 2$$?

Accepted Answer

The graph crosses the x-axis at $$x = 2.$$

Question 4

Which of the following is the correct end behavior for the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

Accepted Answer

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

Question 5

Using the Rational Zeros Theorem, which of the following is a possible rational zero for $$f(x) = 3x^3$ - 5x^2$ + 2x - 8$$?

Accepted Answer

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

Question 6

A rational function $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}$$ has which of the following features?

Accepted Answer

Vertical asymptotes at $$x = 3$$ and $$x = -2$$

Question 7

Given $$f(x) = \frac{2x+1}{x-3}$$, what is the domain of $$f(x)$$?

Accepted Answer

All real numbers except $$x = 3$$

Question 8

Which of the following describes the horizontal asymptote of $$f(x) = \frac{5x^2$ + 3x}{2x^2$ - x + 1}$$?

Accepted Answer

$$y = \frac{5}{2}$$

Question 9

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

Accepted Answer

$$y = 2$$

Question 10

If $$f(x) = x^2$ - 6x + 9$$, use the quadratic formula to find the zeros.

Accepted Answer

$$x = 3$$

Question 11

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

Accepted Answer

$$(x + 2i)(x - 2i)$$

Question 12

Using synthetic division, what is the remainder when $$f(x) = 2x^3$ - 3x^2$ + x - 5 is $$divided by $$x - 2$$?

Accepted Answer

$$1$$

Question 13

Descartes’ Rule of Signs: How many positive real zeros does $$f(x) = x^3$ - 2x^2$ + 3x - 4$$ possibly have?

Accepted Answer

3 or 1

Question 14

A polynomial $$f(x)$$ has degree 5. What is the maximum number of turning points it can have?

Accepted Answer

4

Question 15

The Intermediate Value Theorem states that if $$f(x) is $$continuous on $$[a, b]$$ and $$f(a)$$ and $$f(b)$$ have opposite signs, then:

Accepted Answer

There is at least one zero between $$a$$ and $$b$$

Question 16

Given the zeros $$x = -1$$, $$x = 2$$, and $$x = 3$$ for a cubic polynomial, which of the following is the correct function (up to a constant multiple)?

Accepted Answer

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

Question 17

A medical researcher models the spread of a virus with $$f(x) = \frac{100}{x - 5}$$, where $$x is $$days since exposure. What is the vertical asymptote, and what does it represent?

Accepted Answer

Vertical asymptote at $$x = 5$$, representing the day the model predicts infinite cases.

Question 18

A chemical reaction is modeled by $$f(x) = x^2$ - 2x + 5. $$Using the quadratic formula, what are the zeros of $$f(x)$$?

Accepted Answer

$$x = 1 \pm 2i$$

Question 19

A rational function $$f(x) = \frac{3x^2$ + 2x - 1}{x^2$ - 4} is $$used to model population growth. What is the $$y-$$intercept of the function?

Accepted Answer

$$y = -1$$

Precalculus Study Guide: Piecewise, Polynomial, and Rational Functions

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