Practice 2 Solutions

Name the type of function and describe the transformation from its parent function.

$f (x) = 9 {(x - 2)}^{2}$

Quadratic, $a = 9, h = 2, k = 0$

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty),$ Range: $\{y | y \in ℝ, y \geq 0\}$ or $[0, + \infty)$

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

The quadratic function is stretched vertically by a factor of 9 and translates right 2 as compared to the parent function.

$g (x) = - \sqrt{x - 13} - 7$

Square root, $a = - 1, h = 13, k = - 7$

Domain: $\{x | x \in ℝ, x \geq 13\}$ or $[13, + \infty)$ , Range: $\{y | y \in ℝ, y \leq - 7\}$ or $(- \infty, - 7]$

End behavior: As $x \to 13, g (x) \to - 7$ , and as $x \to \infty, g (x) \to - \infty$

The square root function is reflected over the x-axis and translates right 13 spaces and down 7 spaces as compared to the parent function.

$j (x) = - \frac{6}{5} |x + 1| - 2$

Absolute Value, $a = - \frac{6}{5}, h = - 1, k = - 2$

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$ , Range: $\{y | y \in ℝ, y \leq - 2\}$ or $(- \infty, - 2]$

End behavior: As $x \to + \infty, j (x) \to - \infty$ , and as $x \to - \infty, j (x) \to - \infty$

The absolute value function is stretched by vertically by a factor of $\frac{6}{5}$ and translates 1 space left and down 2 spaces as compared to the parent function.

Write an equation to represent the transformation from a graph. Then describe the end behavior.

Square root

a = 1
h = 0
k = −3

Domain: $\{x | x \in ℝ, x \geq 0\}$ or $[0, + \infty)$

Range: $\{y | y \in ℝ, y \geq - 3\}$ or $[- 3, + \infty)$

$f (x) = \sqrt{x} - 3$

End behavior: As $x \to 0, f (x) \to - 3$ , and as $x \to + \infty, f (x) \to + \infty$

Absolute value

a = 3
h = 4
k = 0

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$

Range: $\{y | y \in ℝ, y \geq 0\}$ or $[0, + \infty)$

$f (x) = 3 |x - 4|$

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

Quadratic

a = −4
h = 1
k = 1

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$

Range: $\{y | y \in ℝ, y \leq 1\}$ or $(- \infty, 1]$

$f (x) = - 4 {(x - 1)}^{2} - 1$

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

Write an equation to represent the transformation from a graph. Name the domain and range in set-builder and interval notation.

Cubic

a = −1
h = 0
k = 2

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

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

Domain: As $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$

Range: $\{y | y \in ℝ\}$ or $(- \infty, + \infty)$

Linear

$a = - \frac{1}{3} h = - 2 k = - 1$

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

$f (x) = - \frac{1}{3} (x + 2) - 1$

$f (x) = - \frac{1}{3} x - \frac{5}{3}$

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$

Range: $\{y | y \in ℝ\}$ or $(- \infty, + \infty)$

Cube Root

a = 2
h = 0
k = 0

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

$f (x) = 2 \sqrt[3]{x}$

Domain: $\{x | x \in ℝ\}$ or $(- \infty, + \infty)$

Range: $\{y | y \in ℝ\}$ or $(- \infty, + \infty)$

Write a function given the description of the transformation as compared to the parent function.

A rational function is reflected across the x-axis, translated right 3 spaces and up 1 space as compared to the parent function.

$f (x) = - \frac{1}{x - 3} + 1$

A cubic function is compressed vertically by a factor of $\frac{1}{2}$ , and translated 6 spaces left as compared to the parent function.

$f (x) = \frac{1}{2} {(x + 6)}^{3}$

A square root function is reflected across both the y-axis and the x-axis as compared to the parent function.

$f (x) = - \sqrt{- x}$