Practice 1 Solutions

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

$f (x) = \frac{3}{5} |x + 8| - 12$

Absolute Value: $a = \frac{3}{5}, h = - 8, k = - 12$

Domain: $\{x | x \in ℝ\} or (- \infty, + \infty),$ Range: $\{y | y \in ℝ, y \geq - 12\} or [- 12, + \infty)$

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

The absolute value graph is compressed vertically by $\frac{3}{5}$ , and translates left 8 and down 12 spaces as compared to the parent function.

$g (x) = 4 x^{2} - 6$

Quadratic, $a = 4, h = 0, k = - 6$

Domain: $\{x | x \in ℝ\} or (- \infty, + \infty),$ Range: $\{y | y \in ℝ, y \geq - 6\} or [- 6, + \infty)$

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 4, and translates down 6 spaces. There is not a horizontal translation.

Name the type of function and describe the transformation from its parent function. Then name the domain and range in set-builder notation.

$h (x) = \frac{- 5}{x + 10} + 15, x \neq - 10$

Rational, $a = - 5, h = - 10, k = 15$

End behavior:
Quadrant II: As $x \to - \infty, f (x) \to 15,$ and as $x \to - 10$ (from the negative side), $f (x) \to + \infty$
Quadrant IV: As $x \to + \infty, f (x) \to 15,$ and as $x \to - 10$ (from the positive side), $f (x) \to - \infty$

The rational function is reflected across the x-axis, stretched vertically by a factor of 5, and translated left 10 and up 15 spaces on the coordinate plane. Domain: $\{x | x \in ℝ\}, x \neq - 10,$ Range: $\{y | y \in ℝ, y \neq 15\}$

$m (x) = \frac{1}{8} \sqrt[3]{x + 18} - 14$

Cube root, $a = \frac{1}{8}, h = - 18, k = - 14$

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

The cube root function is compressed vertically by a factor of $\frac{1}{8}$ , and translates left 18 and down 14 spaces on the coordinate plane.

Doman: $\{x | x \in ℝ\}$ , Range: $\{y | y \in ℝ\}$

Name the type of function and the domain and range in interval notation. Then describe the end behavior.

$j (x) = \sqrt{7 - x} + 1$

$a = 1, h = 7, k = 1$

Square root

Domain: $(- \infty, 7]$ , Range: $[1, + \infty)$

End behavior: As $x \to 7, f (x) \to 1$ , and as $x \to - \infty, f (x) \to + \infty$

Note

Q: What is the transformation that is occurring as compared to the parent function?
A: The square root function is reflected over the y-axis and translated right 7 and up 1 space on the coordinate plane.

$f (x) = 8 {(x + 14)}^{3} - 18$

$a = 8, h = - 14, k = - 18$

Cubic

Domain: $(- \infty, + \infty)$ , Range: $(- \infty, + \infty)$

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

Note

Q: What is the transformation that is occurring as compared to the parent function?
A: The cubic graph is stretched vertically by a factor 8, and translates left 14 and down 18 spaces on the coordinate plane.

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

Quadratic

a = 2
h = 4
k = 1

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

$f (x) = 3 {(x - 4)}^{2} + 1$

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

Range: $\{y | y \in ℝ, y \geq 1\}$ or $[1, + \infty)$

Cubic

a = 5
h = 2
k = 1

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

$f (x) = 5 {(x - 2)}^{3} + 1$

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

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

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

Absolute value

$a = - \frac{1}{2} h = - 5 k = 0$

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

Range: $\{y | y \in ℝ, y \leq 0\}$ or $(- \infty, 0]$

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

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

Square root

This graph is reflected across the y-axis

a = 1
h = 3
k = −4

Domain: $\{x | x \in ℝ, x \leq 3\}$ or $(- \infty, 3]$

Range: $\{y | y \in ℝ, y \geq - 4\}$ or $[4, + \infty)$

$f (x) = \sqrt{3 - x} - 4$

End behavior: As $x \to 3, f (x) \to - 4$ , and as $x \to - \infty, f (x) \to + \infty$

Note

Q: What type of reflection does this graph contain?
A: A reflection over the y-axis.

Q: What indicates this type of reflection?
A: The start/end point is on the right instead of the left of the graph. Also the end behavior shows that x is moving toward negative infinity.

Linear

$a = - \frac{1}{4} h = 2 k = 5$

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

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

$f (x) = - \frac{1}{4} (x - 2) + 5$ or

$f (x) = - \frac{1}{4} x + \frac{11}{2}$

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

Note

You can write the equation in either vertex or slope-intercept form.

Worked solution content here

Cube root

a = 3
h = −1
k = 1

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

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

$f (x) = 3 \sqrt[3]{x + 1} + 1$

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

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

A square root graph is reflected across the x-axis, compressed vertically by a factor of one-third, and translated right 6 spaces and up 9 spaces as compared to the parent graph.

$f (x) = - \frac{1}{3} \sqrt{x - 6} + 9$

An absolute value function is stretched vertically by a factor of 2 and translates 5 spaces left as compared to the parent function.

$f (x) = 2 |x + 5|$

A cubic graph is translated left 7 spaces and down 3 spaces on the coordinate plane as compared to the parent graph.

$f (x) = {(x + 7)}^{3} - 3$