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  5. Rationals - Transformations

Rationals - Transformations

There are various transformations that can be applied to a parent reciprocal function in order to change its core attributes. A transformed function can be expressed algebraically as:

\(f(x) = \cfrac{a}{k(x - d)} + c\)

  • \(a\) represents the vertical stretch/compression factor
  • \(k\) represents the horizontal stretch/compression factor
  • \(d\) represents the horizontal shift and Vertical Asymptote
  • \(c\) represents the vertical shift and Horizontal Asymptote

Vertical Stretch/Compression

The value of \(a\) will determine whether the transformed function will be vertically stretched or compressed:

  • If \(a > 1\), the function will be vertically stretched by a factor of \(a\)
  • If \(0 < a < 1\), the function will be compressed by a factor of \(a\)

\(a > 0\)

Parent Reciprocal function graphed against vertically stretched Reciprocal function.

\(0 < a < 1\)

Parent Reciprocal function graphed against vertically compressed Reciprocal function.

Horizontal Stretch/Compression

The value of \(k\) will determine whether the transformed function will be horizontally stretched or compressed:

  • If \(0 < k < 1\), the function will be horizontally stretched by a factor of \(1/k\)
  • If \(k > 1\), the function will be horizontally compressed by a factor of \(1/k\)

\(0 < k < 1\)

Parent Reciprocal function graphed against horizontally stretched Reciprocal function.

\(k > 1\)

Parent Reciprocal function graphed against horizontally compressed Reciprocal function.

NOTE: With Rational Functions, Compressions and Stretches have the same effect regardless of if they're Horizontal or Vertical. The function's stretch or compression factor depends on the ratio of \(\cfrac{a}{k}\).

Horizontal Shift

The value of \(d\) will determine whether the transformed function will be shifted left or right:

  • If \(d > 0\), the function will be shifted right
  • If \(d < 0\), the function will be shifted left

\(d > 0\)

Parent Reciprocal function graphed against a Reciprocal function shifted to the right.

\(d < 0\)

Parent Reciprocal function graphed against a Reciprocal function shifted to the left.

Vertical Shift

The value of \(c\) will determine whether the transformed function will be shifted upward or downward:

  • If \(c > 0\), the function will be shifted upward
  • If \(c < 0\), the function will be shifted downward

\(c > 0\)

Parent Reciprocal function graphed against a Reciprocal function shifted upwards.

\(c < 0\)

Parent Reciprocal function graphed against a Reciprocal function shifted downward.

Reflections

  • If \(a < 0\), the function will either be vertically stretched or compressed with a reflection in the \(x\)-axis
  • If \(k < 0\), the function will either be horizontally stretched or compressed with a reflection in the \(y\)-axis

\(a < 0\)

Parent Reciprocal function graphed against a Reciprocal function reflected in the x-axis.

\(k < 0\)

Parent Reciprocal function graphed against a Reciprocal function reflected in the y-axis.

Example

For the function \(f(x) = \cfrac{-1}{2x-4} + 6\):

  1. Identify the transformations that were applied
  2. Identify the asymptotes
  3. Identify the domain and range
  4. Sketch the transformed function

i. After shifting some of the values in the transformed equation, we can represent it as:

\(f(x) = \cfrac{-1}{2(x-2)}+ 6\)

We can now determine the transformations more easily:

  • \(\boldsymbol{a = -1}\); reflection in the \(x\)-axis
  • \(\boldsymbol{k = 2}\); horizontal compression by a factor of \(1/2\)
  • \(\boldsymbol{d = 2}\); shifted right by \(2\) units
  • \(\boldsymbol{c = 6}\); shifted upwards \(6\) units

ii. We can identify the asymptotes as such:

  • Since \(c = 6\), we can determine the Horizontal Asymptote (or HA) as \(\boldsymbol{y = 6}\)
  • Since \(d = 2\), we can determine the Vertical Asymptote (or VA) as \(\boldsymbol{x = 2}\)

iii. We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R} | x \neq 2\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R} | y \neq 6\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed Reciprocal function with equation f(x)=-1/[2(x-2)]+6

For the function \(g(x) = \cfrac{5}{x+3}-4\):

  1. Identify the transformations that were applied
  2. Identify the asymptotes
  3. Identify the domain and range
  4. Sketch the transformed function

i. We can now determine the transformations more easily:

  • \(\boldsymbol{a = 5}\); vertical stretch by a factor of \(5\)
  • \(\boldsymbol{d = -3}\); shifted left by \(3\) units
  • \(\boldsymbol{c = -4}\); shifted downward \(4\) units

ii. We can identify the asymptotes as such:

  • Since \(c = -4\), we can determine the Horizontal Asymptote (or HA) as \(\boldsymbol{y = -4}\)
  • Since \(d = -3\), we can determine the Vertical Asymptote (or VA) as \(\boldsymbol{x = -3}\)

iii. We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R} | x \neq -4\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R} | y \neq -3\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed Reciprocal function with equation g(x)=5/(x+3)-4
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