Practice 2 Solutions

1. $f\left(x\right)=9{\left(x–2\right)}^2$

Quadratic, $a=9, h=2, k=0$ 

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right),$ Range: $\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\}$ or $\left[0, +\infty \right)$

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

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

2. $g\left(x\right)=–\sqrt{x–13}–7$

Square root, $a=-1, h=13, k=-7$

Domain: $\left\{x|x\in \mathrm{ℝ}, x\ge 13\right\}$ or $\left[13, +\infty \right)$, Range: $\left\{y|y\in \mathrm{ℝ}, y\le –7\right\}$ or $\left(–\infty , –7\right]$

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

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.

3. $j\left(x\right)=–\frac{6}{5}\left|x+1\right|–2$

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

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$, Range: $\left\{y|y\in \mathrm{ℝ}, y\le –2\right\}$ or $\left(–\infty , –2\right]$

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

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.

Square root

a = 1
h = 0
k = −3

Domain: $\left\{x|x\in \mathrm{ℝ}, x\ge 0\right\}$ or $\left[0, +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}, y\ge –3\right\}$ or $\left[–3, +\infty \right)$

$f\left(x\right)=\sqrt{x}–3$

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

Absolute value

a = 3
h = 4
k = 0

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}, y\ge 0\right\}$ or $\left[0, +\infty \right)$

$f\left(x\right)=3\left|x–4\right|$

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

Quadratic

a = −4
h = 1
k = 1

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}, y\le 1\right\}$ or $\left(–\infty , 1\right]$

$f\left(x\right)=–4{\left(x–1\right)}^2–1$

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

Cubic

a = −1
h = 0
k = 2

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

$f\left(x\right)=–x^3+2$

Domain: As $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Linear

$$\begin{gathered} a=–\frac{1}{3} \\ h=–2 \\ k=–1 \end{gathered}$$

End Behavior: As $x\to +\infty , f\left(x\right)\to –\infty ,$ and as $x\to –\infty ,\mathit{ }f\left(x\right)\to +\infty$

$f\left(x\right)=–\frac{1}{3}\left(x+2\right)–1$

or

$f\left(x\right)=–\frac{1}{3}x–\frac{5}{3}$

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Cube Root

a = 2
h = 0
k = 0

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

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

Domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

Range: $\left\{y|y\in \mathrm{ℝ}\right\}$ or $\left(–\infty , +\infty \right)$

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

$f\left(x\right)=–\frac{1}{x–3}+1$

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

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

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

$f\left(x\right)=–\sqrt{–x}$