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OCF.02.5 – Properties of Reciprocal Functions

OCF.02.5 – Properties of Reciprocal Functions. MCR3U - Santowski. 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 t he time taken to complete a job and the resultant hourly rate

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OCF.02.5 – Properties of Reciprocal Functions

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

  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

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

  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. (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. (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. (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. (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. (C) Graphing f(x) = 2x - 5 and Its Reciprocal • Ex 3 y = 2x - 5 and y = 1/(2x - 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. (C) Graphing f(x) = 0.25x - 3 and Its Reciprocal Ex 4  Graph y = 1/(0.25x – 3) by first graphing y = 0.25x - 3 to identify the asymptotes (x intercept) and the y intercept and two points on either side of the asymptote

  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 = 2f(3x)

  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. (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

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