1. The Reciprocal Function Family.
What you’ll learn
To graph reciprocal functions.
To graph translations of reciprocal functions.
Vocabulary
Reciprocal function, branch.

2. Functions that model inverse variations have the form 𝑓 𝑥 = 𝑎 𝑥 where 𝑥≠0 .
They belong to a family whose parent is the reciprocal function 𝑓 𝑥 = 1 𝑥 where 𝑥≠0
Essential Understanding Transformations of the parent reciprocal function include stretches, compressions or shrinks, reflections, and horizontal and vertical translations.

3. Problem 1: Graphing an inverse Variation Function
What is the graph of 𝑦= 8 𝑥 ,𝑥≠0? Identify the x-and y-intercept and the asymptotes of the graph. Also, state the domain and the range of the function.
Step1: Make a table of values of x (positive and negative). x y
-16
-1/2 -8 -1 -4 -2 x y
1/2 16 1 8 2 4
You can use the G.C. and get the
ordered pairs from the table.

4. Step 3:Connect the points with a smooth curve.
Step2: Graph the points
Step 3:Connect the points with a smooth curve.
x cannot be zero, so there is not y-intercept.
The numerator is never zero, so y is never 0.
There is no x-intercept.
The x- axis is horizontal asymptote.
The y-axis is vertical asymptote.
Domain: all real numbers except x=0.
Range: all real numbers except y=0

5. Your Turn
What is the graph of y= 12 x ? Identify the x-and y-intercept and the asymptotes of the graph. Also, state the domain and the range of the function.
Reasoning Would the function 𝑦= 6 𝑥 have the same domain and range as y= 8 x or 𝑦= 12 𝑥 ? Explain
Answer:

6. Problem 2:Identifying reciprocal Function transformations.
For each value of a, how do the graph of 𝑦= 1 𝑥 𝑎𝑛𝑑 𝑦= 𝑎 𝑥 compare?What is the effect of a on the graph?
Answer

7. You can translate any reciprocal function horizontally or vertically just as you can other functions. 

8. Problem 3: Graphing a Transformation.
What is the graph of 𝑦= 1 𝑥+1 −2 ?Identify domain and range.
Answer
Step 1: Draw the asymptotes.
The Vertical asymptote is 𝑥=−1
The Horizontal asymptote is y=−2
Step 2: Translate the graph of 𝑦= 1 𝑥
The graph 𝑦= 1 𝑥 contains the points (1, 1) and (-1, -1 ) Translate these points 1 unit to the left and 2 units down. Draw the branches through these points.
Domain: is the set of all real numbers except x=-1. The range is the set of all real numbers except y=-2

9. Your turn.
What is the graph of y= 1 x−3 +2 ? Identify domain and range
Answer
The Vertical asymptote is 𝑥=3
y=2
The Horizontal asymptote is y=2
x=3

10. Answer: Answer: Problem 4: Writing the equation of a transformation.
This graph of a function is a translation of y= 2 x What is the equation for the function?
Answer:
asymptotes are 𝑥=−3 𝑎𝑛𝑑 𝑦=4
therefore h=−3 𝑎𝑛𝑑 𝑘=4
𝑦= 𝑎 𝑥−ℎ +𝑘
Use the general form
y= 2 x−(−3) +4
Substitute for a, h, and k
𝑦= 2 𝑥+3 +4
Simplify
Your turn.
Answer:
This graph of a function is a translation
of 𝑦= 2 𝑥 What is the equation for the function?