Algebra II
9.2 The Reciprocal Function Family
LEQ: How do you graph reciprocal functions?
Procedure:
1. Using Graphs of Inverse Variations:
a. Definition 1: Functions that model inverse variations belong to a family
whose parent is the ______________________________
b. Definition 2: The graph below shows the function y = 5/x. Notice that the
graph has two parts. Each part is called a ______________.
c. Definition 3: The _____________________________ of a graph are
lines the graph approaches.
i. What is the equation of the vertical asymptote?
ii. What is the equation of the horizontal asymptote?
iii. Graph y = 1/x. Sketch the graph. Clear your screen and graph
y = -1/x. Sketch the graph. Compare the two sketches. How does
the negative sign affect the graph? Describe the asymptotes of the
two graphs.
iv. Graph y = 1/x2. Sketch the graph. Compare the sketch with the
graph of y = 1/x from part iii. How does squaring x affect the graph?
Describe the asymptotes of the two graphs.
d. Example 1: Relating to the Real World:
Music: A musical pitch is measured in vibrations per second, or Hertz
(Hz). The pitch y produced by a panpipe varies inversely with the length of
the pipe x, measured in feet. The equation y = 564/x models the inverse
variation. Find the length of the pipe that produces a pitch of 277 Hz.
i. Pitches of 247 Hz, 311 Hz, and 370 Hz form a musical chord. Find
the length of pipe that will produce each pitch.
ii. The asymptotes of y = 264/x are x = 0 and y = 0. Explain why this
makes sense in terms of the panpipe.
2. Graphing Translations of Inverse Variation:
The graphs below show the function y = 4/x, y = 4/x + 2, and y = 4/x – 4.
i. What is the vertical asymptote of each graph? The horizontal
asymptote?
ii. How are the graphs of y = 4/x and y = 4/x + c related?
The graphs below show the functions y = 4/x, y = 4/(x – 2), and y = 4/(x + 4).
iii. What is the vertical asymptote of each graph? The horizontal
asymptote?
iv. How are the graphs of y = 4/x and y = 4/(x – b) related?
b. Translation of Inverse Variation:
The graph of y = k/(x – b) + c is a translation of y = k/x that has moved b
units horizontally and c unit vertically. It has a vertical asymptote at x = b
and a horizontal asymptote at y = c.
c. Example 2 – 7: Sketching Graphs:
Sketch the graphs of the following functions. Identify the vertical and
horizontal asymptotes.
d. Example 8: Writing Equations:
Write an equation for a translation of y = 5/x that has asymptotes at x = -2
and y = 3.
3. HW: pg. 498 (2 – 24 evens, 42 – 46 evens)