Practice 2 Solutions

Match the given graph to its end behavior. Choices may be used more than once.

As $x \to + \infty, f (x) \to + \infty,$ and as $x \to - \infty, f (x) \to - \infty$
As $x \to + \infty, f (x) \to + \infty,$ and as $x \to 0, f (x) \to 0$
As $x \to + \infty, f (x) \to + \infty,$ and as $x \to - \infty, f (x) \to + \infty$
Quadrant I: As $x \to + \infty, f (x) \to 0,$ and as $x \to 0, f (x) \to + \infty$
Quadrant III: As $x \to - \infty, f (x) \to 0,$ and as $x \to 0, f (x) \to - \infty$

A 1)

B 2)

C 3)

D 4)

Name the parent function in given graphs. Then name the domain and range.

Graph of problem 1

cube root $y = \sqrt[3]{x}$

$domain : \{x | x \in ℝ\} or (- \infty, \infty) range : \{y | y \in ℝ\} or (- \infty, \infty)$

Graph of problem 2

square root $y = \sqrt{x}$

$domain : \{x | x \in ℝ, x \geq 0\} or [0, \infty) range : \{y | y \in ℝ, y \geq 0\} or [0, \infty)$

Graph of problem 3

absolute value $y = |x|$

$domain : \{x | x \in ℝ\} or (- \infty, \infty) range : \{y | y \in ℝ, y \geq 0\} or [0, \infty)$

Graph of problem 4

rational $y = \frac{1}{x}, x \neq 0$

$domain : \{x | x \in ℝ, x \neq 0\} range : \{y | y \in ℝ, y \neq 0\}$

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

Note

This problem is an even degree polynomial, and its end behavior is in the same direction.

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

Note

This problem is an odd degree polynomial, and its end behavior is in opposite directions.

Given the equation of the parent function, sketch the graph. Then name the end behavior.

Note

Problems 11–13

Q: Which are polynomial functions? Which are not?
A: All are polynomial functions.

quadratic

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

cubic

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

quadratic

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

Describe the end behavior of a polynomial parent function when the degree is odd and when the degree is even.

The end behavior for an odd degree polynomial parent graph will be: As x approaches positive infinity, $f (x)$ approaches positive infinity, and as x approaches negative infinity, $f (x)$ approaches negative infinity. The end behavior for an even degree polynomial parent graph will be: As x approaches positive infinity, $f (x)$ approaches positive infinity, and as x approaches negative infinity, $f (x)$ approaches positive infinity.