Explore: Domain, Range, and End Behavior of Transformations Solutions

Domain, Range, and End Behavior of Transformations Solutions

The domain and range of transformed functions may change depending on the value of a, h, and k.
The domain will be all real numbers ${x | x \in ℝ}$ unless there is an excluded value or restriction for the function.
The range will depend on whether a function has a defined point or vertex or neither .
The leading coefficient , a, is the only value that will affect the end behavior of a function.

Name the domain and range in set-builder notation. Then name the end behavior.

$h (x) = - 2 {(x - 6)}^{2} + 3$

Type of function: quadratic

Identify a, h, k: $a = - 2, h = 6, k = 3$

Domain: $\{x | x \in ℝ\}$

Range: $\{y | y \in ℝ, y \leq 3\}$

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

Note

The k-value of the parabola’s vertex $(h, k)$ determines the range value.

Write the equation of the given function. Then name the domain and range in set-builder notation and the end behavior.

Square root

$a = - 1, h = - 3, k = 4 y = - \sqrt{x + 3} + 4$

domain: $\{x | x \in ℝ, x \geq - 3\}$

range: $\{y | y \in ℝ, y \leq 4\}$

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

Note

For the square root function, $(h, k)$ determines the start/end point and what x and $f (x)$ approach for the end behavior.

Sketch the transformation from the parent graph. Then name the domain and range in set-builder notation and the end behavior.

$g (x) = \frac{1}{2} \sqrt[3]{x + 1} - 1$

domain: $\{x | x \in ℝ\}$

range: $\{y | y \in ℝ\}$

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

Given the equation, name the domain and range in set-builder notation and the end behavior.

$f (x) = |x + 7| - 10$

Absolute value

$a = 1, h = - 7, k = - 10$

domain: $\{x | x \in ℝ\}$

range: $\{y | y \in ℝ, y \geq - 10\}$

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