Mastery Check Solutions

Show What You Know

A reciprocal function is stretched by 3 with the domain $\{x | x \in ℝ, x \neq - 4\}$ and a range of $\{y | y \in ℝ, y \neq 2\}$ .

Write an equation for a reciprocal function in the form:
$y = \frac{1}{x - h} + k, x \neq h$ .

$a = 3, h = - 4, k = 2 y = \frac{3}{x + 4} + 2, x \neq - 4$

Part B intercepts:

$\begin{array}{rcl} 0 & = & \frac{3}{x + 4} + 2 \\ - 2 & = & \frac{3}{x + 4} \\ - 2 (x + 4) & = & 3 \\ - 2 x - 8 & = & 3 \\ - 2 x & = & 11 \\ x & = & - \frac{11}{2} \\ (- 5.5, 0) \end{array}$

$\begin{array}{rcl} y & = & \frac{3}{0 + 4} + 2 \\ y & = & \frac{3}{4} + 2 \\ y & = & 2 \frac{3}{4} \\ (0, 2.75) \end{array}$

Note

Q: What variable determines the vertical stretch of a function? What is it for this problem?

A: a = 3

Q: Does this graph have any horizontal and/or vertical shifts? Explain.
A: Yes, since h = –4, the graph shifts left 4 spaces. Because k = 2, the graph shifts up 2 spaces compared to the parent function.

Graph the equation from part A. Label asymptotes and intercepts.

Note

Q: In what quadrant do the asymptotes intersect?

A: Quadrant II

Use the numbers {–3, –2, –1, 0, 1, 2} only once to create a reciprocal function with the following characteristics:

- Be a reflection with no stretch or compression 
- The vertical and horizontal asymptotes must intersect in Quadrant IV

Write the equation of the reciprocal function.
a = –1, h = 1 or 2, k = –2 or –3

$For a = - 1, h = 1, k = - 2 f (x) = \frac{- 1}{x - 1} + (- 2)$

$For a = - 1, h = 2, k = - 2 f (x) = \frac{- 1}{x - 2} + (- 2)$

$For a = - 1, h = 1, k = - 3 f (x) = \frac{- 1}{x - 1} + (- 3)$

$For a = - 1, h = 2, k = - 3 f (x) = \frac{- 1}{x - 2} + (- 3)$

Name the domain and range for the equation in part C.
$For a = - 1, h = 1, k = - 2 Domain : \{x | x \in ℝ, x \neq 1\}} Range : \{y | y \in ℝ, y \neq - 2\}}$

$For a = - 1, h = 2, k = - 2 Domain : \{x | x \in ℝ, x \neq 2\}} Range : \{y | y \in ℝ, y \neq - 2\}}$

$For a = - 1, h = 1, k = - 3 Domain : \{x | x \in ℝ, x \neq 1\} Range : \{y | y \in ℝ, y \neq - 3\}}$

$For a = - 1, h = 2, k = - 3 Domain : \{x | x \in ℝ, x \neq 2\}} Range : \{y | y \in ℝ, y \neq - 3\}}$

Note

There are four possible equations using the given values. You should keep working until you have at least one correct equation. Rather than erasing equations that do not follow the guidelines, put an “x” next to the equation so you can see what is working and what is not working.

You can use More to Explore to see how to use technology to check your work and confirm a correct equation.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page. 

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

- Determine the vertical and horizontal asymptotes for a reciprocal function.
- Determine the transformations of a graph from the reciprocal parent function.
- Graph/sketch a reciprocal function in the form of: $y = \frac{a}{x - h} + k$
- State the domain and range for a reciprocal function.