The Reciprocal Function Family and rational functions and their graphs Lesson 9-2 and Lesson 9-3

Functions that model inverse variations belong to a family whose parent is the reciprocal function 𝑓 𝑥 = 1 𝑥 , where 𝑥≠0. Transformations 𝑓 𝑥 = 𝑎 𝑥 +𝑘 𝑚𝑜𝑣𝑒𝑠 𝑢𝑝 𝑓 𝑥 = 𝑎 𝑥 −𝑘 𝑚𝑜𝑣𝑒𝑠 𝑑𝑜𝑤𝑛 𝑓 𝑥 = 𝑎 𝑥−ℎ 𝑚𝑜𝑣𝑒𝑠 𝑟𝑖𝑔ℎ𝑡 𝑓 𝑥 = 𝑎 𝑥+ℎ 𝑚𝑜𝑣𝑒𝑠 𝑙𝑒𝑓𝑡 a is the stretch (if 𝑎>1) 𝑜𝑟 𝑠ℎ𝑟𝑖𝑛𝑘 𝑖𝑓 0<𝑎<1 𝑎<0 is a reflection in the x-axis

Graphing an Inverse Variation Sketch a graph of 𝑦= 6 𝑥 , 𝑥≠0 What are the asymptotes? Each part of the graph is called a branch.

Graphing Reciprocal Functions Draw the graph of 𝑦= −4 𝑥 Describe the transformations

Real-world connection A musical pitch is determined by the frequency of vibration of the sound waves reaching the ear. The greater the frequency, the higher is the pitch. Frequency is measured in vibrations per second, or hertz (Hz). The pitch (y) produced by a panpipe varies inversely with the length (x) of the pipe. Write the function: ___________________ Find the length of the pipe that produces a pitch of 277 Hz. Pitches of 247 Hz and 370 Hz. Find the length of pipes that will produce each pitch. The asymptotes of this equation are y=0 and x=0. Explain why this makes sense in terms of the panpipe. Desmos Graphing Calculator

Graphing translations of reciprocal functions Graph on desmos 𝑦= 1 𝑥 , y= 1 𝑥−1 , 𝑦= 1 𝑥+2 What are the vertical and horizontal asymptotes for each graph? How do the vertical asymptotes relate to the denominators equaling zero? Now graph 𝑦= 1 𝑥 , 𝑦= 1 𝑥 +1, 𝑎𝑛𝑑 𝑦= 1 𝑥 −2 What are the vertical and horizontal asymptotes for each graph?

Graphing a translation Sketch the graph of 𝑦= 1 𝑥−2 −3

Writing the equation of a transformation Write an equation for the translation of 𝑦= 5 𝑥 that has asymptotes at 𝑥=−2 𝑎𝑛𝑑 𝑦=3. Write an equation for the translation of 𝑦= −1 𝑥 that is 4 units left and 5 units up. Check your work by graphing your solution.

L9-3 Objective: Students will identify properties of rational functions

A point of discontinuity is either a hole or a vertical asymptote. If it makes the denominator equal zero but does not make the numerator equal zero it is an asymptote. If it makes both the denominator and the numerator equal zero it is a hole.

For each rational function, find any points of discontinuity. Rational Functions and Their Graphs LESSON 9-3 Additional Examples For each rational function, find any points of discontinuity. a. y = 3 x2 – x –12 The function is undefined at values of x for which x2 – x – 12 = 0. x2 – x – 12 = 0 Set the denominator equal to zero. (x – 4)(x + 3) = 0 Solve by factoring or using the Quadratic Formula. x – 4 = 0 or x + 3 = 0 Zero-Product Property x = 4 or x = –3 Solve for x. There are points of discontinuity at x = 4 and x = –3.

Since –1 and –5 are the zeros of the denominator and neither is a zero Rational Functions and Their Graphs LESSON 9-3 Additional Examples Describe the vertical asymptotes and holes for the graph of each rational function. a. y = x – 7 (x + 1)(x + 5) Since –1 and –5 are the zeros of the denominator and neither is a zero of the numerator, x = –1 and x = –5 are vertical asymptotes. b. y = (x + 3)x x + 3 –3 is a zero of both the numerator and the denominator. The graph of this function is the same as the graph y = x, except it has a hole at x = –3. c. y = (x – 6)(x + 9) (x + 9)(x + 9)(x – 6) 6 is a zero of both the numerator and the denominator. The graph of the function is the same as the graph y = which has a vertical asymptote at x = –9, except it has a hole at x = 6. 1 (x + 9) ,

Homework: L9-2 (506) #2, 6, 16, 22, 24, 28, 30, 32 L9-3 (p513) #1-18