Reciprocal Transformations (DP IB Analysis & Approaches (AA)): Revision Note

Reciprocal transformations

What effects do reciprocal transformations have on the graphs?

  • The x-coordinates stay the same

  • The y-coordinates change

    • Their values become their reciprocals

  • The coordinates (x, y) become open parentheses x comma 1 over y close parentheses where y ≠ 0

    • If y = 0 then a vertical asymptote goes through the original coordinate

    • Points that lie on the line y = 1 or the line y = -1 stay the same

How do I sketch the graph of the reciprocal of a function: y = 1/f(x)?

  • Sketch the reciprocal transformation by considering the different features of the original graph

  • Consider key points on the original graph

    • If (x1, y1) is a point on y = f(x) where y1 ≠ 0

      • open parentheses x subscript 1 comma 1 over y subscript 1 close parentheses is a point on y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction

      • If |y1| < 1 then the point gets further away from the x-axis

      • If |y1| > 1 then the point gets closer to the x-axis

    • If y = f(x) has a y-intercept at (0, c) where c ≠ 0

      • The reciprocal graph y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a y-­intercept at open parentheses 0 comma 1 over c close parentheses

    • If y = f(x) has a root at (a, 0)

      • The reciprocal graph y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a vertical asymptote at x equals a

    • If y = f(x) has a vertical asymptote at x equals a

      • The reciprocal graph y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a discontinuity at (a, 0) 

      • The discontinuity will look like a root

    • If y = f(x) has a local maximum at (x1, y1) where y1 ≠ 0

      • The reciprocal graph y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a local minimum at open parentheses x subscript 1 comma 1 over y subscript 1 close parentheses

    • If y = f(x) has a local minimum at (x1, y1) where y1 ≠ 0

      • The reciprocal graph y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a local maximum at open parentheses x subscript 1 comma 1 over y subscript 1 close parentheses

  • Consider key regions on the original graph

    • If y = f(x) is positive then y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction is positive

      • If y = f(x) is negative then y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction is negative

    • If y = f(x) is increasing then y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction is decreasing

      • If y = f(x) is decreasing then y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction is increasing

    • If y = f(x) has a horizontal asymptote at y =

      • y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a horizontal asymptote at y equals 1 over k if k ≠ 0

      • y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction tends to ± ∞ if k = 0

    • If y = f(x) tends to ± ∞ as tends to +∞ or -∞

      • y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction has a horizontal asymptote at y equals 0

Worked Example

The diagram below shows the graph of y equals f left parenthesis x right parenthesis which has a local maximum at the point A.

2-9-2-we-image

Sketch the graph of .y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction.

2-9-2-ib-aa-hl-reciprocal-trans-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.