IBMaths: AA HLDPRevision Notes2. Functions2.9 Further Functions & Graphs2.9.4 Reciprocal & Square Transformations

Reciprocal & Square Transformations (DP IB Maths: AA HL)

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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 = k 

      • 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 x 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

Square Transformations

What effects do square transformations have on the graphs?

  • The effects are similar to the transformation y = |f(x)|
    • The parts below the x-axis are reflected
    • The vertical distance between a point and the x-axis is squared
      • This has the effect of smoothing the curve at the x-axis
  • y equals open square brackets f open parentheses x close parentheses close square brackets squared is never below the x-axis
  • The x-coordinates stay the same
  • The y-coordinates change
    • Their values are squared
  • The coordinates (x, y) become (x, y²)
    • Points that lie on the x-axis or the line y = 1 stay the same

How do I sketch the graph of the square of a function: y = [f(x)]²?

  • Sketch the square 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)
      •  open parentheses x subscript 1 comma y subscript 1 squared close parentheses is a point on y equals open square brackets f open parentheses x close parentheses close square brackets squared
      • If |y1| < 1 then the point gets closer to the x-axis
      • If |y1| > 1 then the point gets further away from the x-axis
    • If y = f(x) has a y-intercept at (0, c)
      • The square graph y equals open square brackets f open parentheses x close parentheses close square brackets squared has a y-­intercept at left parenthesis 0 comma c squared right parenthesis
    • If y = f(x) has a root at (a, 0)
      • The square graph  y equals open square brackets f open parentheses x close parentheses close square brackets squared has a root and turning point at (a, 0)
    • If y = f(x) has a vertical asymptote at x equals a
      • The square graph y equals open square brackets f open parentheses x close parentheses close square brackets squared has a vertical asymptote at x equals a
    • If y = f(x) has a local maximum at (x1, y1)
      • The square graph  y equals open square brackets f open parentheses x close parentheses close square brackets squared has a local maximum at (x1, y12) if y1 > 0
      • The square graph y equals open square brackets f open parentheses x close parentheses close square brackets squared  has a local minimum at (x1, y12) if y1 ≤ 0
    • If y = f(x) has a local minimum at (x1, y1)
      • The square graph y equals open square brackets f open parentheses x close parentheses close square brackets squared  has a local minimum at (x1, y12) if y1 ≥ 0
      • The square graph  y equals open square brackets f open parentheses x close parentheses close square brackets squared has a local maximum at (x1, y12) if y1 < 0

Exam Tip

  • In an exam question when sketching y equals open square brackets f left parenthesis x right parenthesis close square brackets squared make it clear that the points where the new graph touches the x-axis are smooth
    • This will make it clear to the examiner that you understand the difference between the roots of the graphs y equals open vertical bar f left parenthesis x right parenthesis close vertical bar and y equals open square brackets f left parenthesis x right parenthesis close square brackets squared

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 open square brackets f open parentheses x close parentheses close square brackets squared.

2-9-2-ib-aa-hl-square-trans-we-solution

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Dan

Author: Dan

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.