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  5. Polynomials - Transformations

Polynomials - Transformations

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

\(f(x) = a[k(x - d)]^n+c\)

  • \(a\) represents the vertical stretch/compression factor.
  • \(k\) represents the horizontal stretch/compression factor
  • \(d\) represents the horizontal shift
  • \(c\) represents the vertical shift
  • \(n\) represents the degree

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
  • If \(0 < a < 1\), the function will be vertically compressed
  • If \(a < 0\), the function will either be vertically stretched or compressed with a reflection in the x-axis

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

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

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

Example

For the function \(f(x) = -3(4x-20)^2+7\):

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

i. We can identify the parent function as \(\boldsymbol{f(x) = x^2}\).

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

\(f(x) = -3[4(x-5)]^2 + 7\)

We can now determine the transformations more easily:

  • \(\boldsymbol{a = -3}\); vertical stretch by a factor of \(3\) and reflection in the \(x\)-axis
  • \(\boldsymbol{k = 4}\); horizontal compression by a factor of \(4\)
  • \(\boldsymbol{d = 5}\); shifted right by \(5\) units
  • \(\boldsymbol{c = 2}\); shifted upwards \(7\) units

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

  • Domain: \(\boldsymbol{\{x\in\mathbb{R}\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R} | y \leq 7\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed quadratic function expressed as f(x)=-3(4(x-5))²+7

For the function \(g(x) = 2(-x-1)^3-6\):

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

i. We can identify the parent function as \(\boldsymbol{g(x) = x^3}\).

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

\(g(x) = \cfrac{1}{2}[-(x+1)]^3-6\)

We can now determine the transformations more easily:

  • \(\boldsymbol{a = 2}\); vertical stretch by a factor of \(2\)
  • \(\boldsymbol{k = -1}\); reflection in the \(y\)-axis
  • \(\boldsymbol{d = -1}\); shifted left by \(1\) unit
  • \(\boldsymbol{c = -6}\); shifted downward \(6\) units

We can identify the domain and range as such:

  • Domain: \(\boldsymbol{\{x\in\mathbb{R}\}}\)
  • Range: \(\boldsymbol{\{y\in\mathbb{R}\}}\)

iv. We can draw our transformed graph as such:

Graph of transformed Cubic Function expressed as g(x)=1/2[-(x+1)]³-6.

Shortcuts for Transforming Functions

Since we've had some experience transforming functions, we can use the following steps to make the process more convenient for sketching the transformed functions:

  1. Factor out \(k\) to see \(d\)
  2. Fill out the 3 tables below
  3. Plot the final table



Example

Write the equation for the following transformed function. Sketch using the shortcut:
\(y = x^5\) reflected in the \(y\)-axis, horizontal stretch by \(2\), horizontal shift up by \(9\) units.

Before writing out our equation, we can determine the transformations:

  • \(k = -1/2\)
  • \(c = 9\)

Now, we can write the equation of the transformed function:

\(y = \left[-\cfrac{1}{2}(x)\right]^5+3\)

Finally, we can use our shortcut tables to help draw our graph:

x -3 -2 -1 0 1 2 3
y (Parent) 243 32 1 0 -1 -32 -243

x ÷ -1/2 6 4 2 0 -2 -4 -6
y 243 32 1 0 -1 -32 -243

x 6 4 2 0 -2 -4 -6
y + 3 246 35 4 3 2 -29 -240

Now, we can draw our transformed graph as such:

Graph of transformed quintic function expressed as y=[1/2(x)]⁵+3.
Write the equation for the following transformed function. Sketch using the shortcut:
\(i(x) = x^4\) reflected in the \(x\)-axis, vertical compression by \(\cfrac{1}{4}\), shift left \(6\) units, translation \(2\) units downward.

Before writing out our equation, we can determine the transformations:

  • \(a = -1/4\)
  • \(d = 6\)
  • \(c = -2\)

Now, we can write the equation of the transformed function:

\(i(x) = -\cfrac{1}{4}(x-6)^4-2\)

Finally, we can use our shortcut tables to help draw our graph:

x -3 -2 -1 0 1 2 3
i(x) (Parent) 81 16 1 0 1 16 81

x -3 -2 -1 0 1 2 3
y . -1/4 -20.25 -4 -0.25 0 -0.25 -4 -20.25

x+6 3 4 5 6 7 8 9
i(x)-2 -22.25 -6 -2.25 -2 -2.25 -6 -22.25

Now, we can draw our transformed graph as such:

Graph of transformed quartic function expressed as i(x)=-1/4(x-6)⁴-2.
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