1. 1 OCF.02.5 – Properties of Reciprocal Functions MCR3U - Santowski

2. 2 Introduction to Reciprocal Functions Consider the idea of being paid $450 to complete a painting job  you have to paint my home and we will compare the time taken to complete a job and the resultant hourly rate Here we observe the idea that as the time to paint my house increases, the hourly rate decreases. Thus we have an inverse variation  as one quantity goes up, the other goes down Mathematically, we can express this as xy = k or on our case (time taken)(hourly rate)= 450 We can rearrange these equation to create new ones: xy = k can become either y = k / x or x = k / y and in our case, (time taken) = 450/(hourly rate) or (hourly rate) = 450/(time taken) so we have the equations t = 450/h or h = 450/t Hours Worked 31015202530455090150 Hourly Rate 150453022.50181510953

3. 3 Introduction to Reciprocal Functions Here is a scatter plot showing the data

4. 4 (B) The Reciprocal Function f(x) = 1/ x We can generate a graph of y = 1/ x using graphing technology and make some observations: (This graph is called a hyperbola)

5. 5 (B) Table of Values for f(x) = 1/x x y -10.00000 -0.10000 -9.00000 -0.11111 -8.00000 -0.12500 -7.00000 -0.14286 -6.00000 -0.16667 -5.00000 -0.20000 -4.00000 -0.25000 -3.00000 -0.33333 -2.00000 -0.50000 -1.00000 -1.00000 0.00000 undefined 1.00000 1.00000 2.00000 0.50000 3.00000 0.33333 4.00000 0.25000 5.00000 0.20000 6.00000 0.16667 7.00000 0.14286 8.00000 0.12500 9.00000 0.11111 10.00000 0.10000

6. 6 (B) Features of The Reciprocal Function f(x) = 1/ x 1. Domain and domain restrictions 2. Range and range restrictions 3. Vertical asymptotes => what happens as x gets closer and closer to 0 (or the domain restriction); (and why). State equation of the vertical asymptote 4. Horizontal asymptotes => what happens as x gets larger and larger (positive and negative); (and why). State equation of the horizontal asymptote 5. Since the asymptotes are perpendicular, this hyperbola is called a rectangular hyperbola 6. Find the inverse of y = 1/ x

7. 7 (C) Graphing f(x) = x and Its Reciprocal (Linear Functions) example 1  f ( x ) = x and y = 1/ x Note the root/zero of f ( x )  (0,0) and what happens to the reciprocal function at that same x value (asymptotes) Note the intersection point of f ( x ) and its reciprocal.

8. 8 (C) Graphing f(x) = x + 1 and Its Reciprocal Ex 2  y = x + 1 and y = 1/( x + 1). Note the root/zero of f ( x )  (0,-1) and what happens to the reciprocal function at that same x value (asymptotes) Note the intersection point of f ( x ) and its reciprocal.

9. 9 (C) Graphing f(x) = 2x - 5 and Its Reciprocal Ex 3  y = 2 x - 5 and y = 1/(2 x - 5) Note the root/zero of f ( x )  (0,2.5) and what happens to the reciprocal function at that same x value (asymptotes) Note the intersection point of f ( x ) and its reciprocal.

10. 10 (C) Graphing f(x) = 0.25x - 3 and Its Reciprocal Ex 4  Graph y = 1/(0.25 x – 3) by first graphing y = 0.25 x - 3 to identify the asymptotes ( x intercept) and the y intercept and two points on either side of the asymptote

11. 11 (E) Graphing Reciprocal Functions by means of Transformations Likewise, we can also pursue transformations of y = 1/ x by moving their asymptotes and several key points: 1. Graph f ( x ) = 1/ x and then make note of the asymptotes and two keys points (1,1) and (-1,-1) 2. Graph y = f ( x + 3) by moving the asymptotes and the key points 3. Repeat for y = - f ( x ) ; y = f (- x ); y = f ( x - 4) + 3 ; y = 2 f (3 x )

12. 12 (F) Internet Links Rational Functions - An Interactive Tutorial from AnalyzeMath Another Tutorial on Rational Functions from AnalyzeMath Graphing Rational Equations Lesson from Purple Math Asymptotes Lesson from PurpleMath Graphs of Rational Functions from West Texas A&M

13. 13 (F) Homework Nelson text, page 345 - 348, Q3ii, 5, 11, 13, 15 Work with inverse variations would be Q1,7,8 Work with transformations would be from Harcourt Math 11, page 28, Q1