Transformations of Functions

Overview

Transformations of functions refer to the various ways in which the graph of a function can be shifted, stretched, compressed, or reflected. These transformations allow us to manipulate the graph of a basic function to fit specific conditions without having to draw the entire function from scratch.

Types of Transformations

  • Translation: Shifting the graph horizontally or vertically.
  • Reflection: Flipping the graph over a line (usually the x-axis or y-axis).
  • Stretching/Compression: Changing the scale of the graph in either the vertical or horizontal direction.

Translation

Translation involves shifting the graph of a function either horizontally or vertically.

  • Vertical Translation: To translate a graph vertically, add or subtract a constant \( k \) from the function. If \( f(x) \) is the original function, then \( f(x) + k \) shifts the graph vertically by \( k \) units. If \( k > 0 \), the graph shifts upwards, and if \( k < 0 \), the graph shifts downwards.
  • Horizontal Translation: To translate a graph horizontally, replace \( x \) with \( x - h \). The graph of \( f(x - h) \) shifts the function \( h \) units to the right if \( h > 0 \), and \( h \) units to the left if \( h < 0 \).

Reflection

Reflection involves flipping the graph over a line.

  • Reflection over the x-axis: To reflect a graph over the x-axis, multiply the function by -1. The equation \( -f(x) \) reflects the graph of \( f(x) \) over the x-axis.
  • Reflection over the y-axis: To reflect a graph over the y-axis, replace \( x \) with \( -x \). The equation \( f(-x) \) reflects the graph of \( f(x) \) over the y-axis.

Stretching and Compression

Stretching and compression change the scale of the graph.

  • Vertical Stretch/Compression: To stretch or compress a graph vertically, multiply the function by a constant \( a \). If \( a > 1 \), the graph is stretched vertically, and if \( 0 < a < 1 \), the graph is compressed vertically.
  • Horizontal Stretch/Compression: To stretch or compress a graph horizontally, replace \( x \) with \( \frac{x}{b} \). If \( b > 1 \), the graph is compressed horizontally, and if \( 0 < b < 1 \), the graph is stretched horizontally.

Combining Transformations

We can combine translations, reflections, and stretches/compressions to create more complex transformations. The general form of a transformed function is:

\[ y = a(f(bx - h)) + k \]

Where:

  • \( a \) stretches or reflects vertically.
  • \( b \) compresses or stretches horizontally.
  • \( h \) shifts the graph horizontally.
  • \( k \) shifts the graph vertically.

Example 1: Vertical and Horizontal Translation

Consider the function \( f(x) = x^2 \). To translate the graph of \( f(x) \) 3 units to the right and 2 units up, the transformed function is:

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

This shifts the graph 3 units to the right and 2 units up.

Example 2: Reflection and Vertical Stretch

For the function \( f(x) = x^2 \), to reflect it over the x-axis and vertically stretch it by a factor of 2, the transformed function is:

\[ f(x) = -2x^2 \]

This reflects the graph over the x-axis and stretches it vertically by a factor of 2.

Practice Questions

  1. Given \( f(x) = x^2 \), what is the transformed function if the graph is shifted 4 units left and 3 units down?
    Solution

    The transformation involves shifting the graph horizontally by 4 units left and vertically by 3 units down. The transformed function is:

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

  2. Given \( f(x) = \sqrt{x} \), what is the transformed function if the graph is stretched vertically by a factor of 3 and reflected over the y-axis?
    Solution

    The vertical stretch by a factor of 3 is achieved by multiplying the function by 3, and the reflection over the y-axis is achieved by replacing \( x \) with \( -x \). The transformed function is:

    \[ f(x) = -3\sqrt{-x} \]

  3. Given \( f(x) = x^3 \), what is the transformed function if the graph is stretched vertically by a factor of 2, compressed horizontally by a factor of 0.5, and shifted 5 units up?
    Solution

    To apply the transformations, multiply the function by 2 for vertical stretch, replace \( x \) with \( 2x \) for horizontal compression by a factor of 0.5, and add 5 for the vertical shift. The transformed function is:

    \[ f(x) = 2(2x)^3 + 5 = 16x^3 + 5 \]

  4. Given \( f(x) = 2x + 1 \), what is the transformed function if the graph is reflected over the x-axis and shifted 3 units to the left and 4 units down?
    Solution

    To reflect over the x-axis, multiply the function by -1, to shift 3 units left replace \( x \) with \( x + 3 \), and to shift 4 units down subtract 4 from the function. The transformed function is:

    \[ f(x) = -2(x + 3) + 1 - 4 = -2(x + 3) - 3 \]

  5. Given \( f(x) = |x| \), what is the transformed function if the graph is reflected over the y-axis and stretched vertically by a factor of 3?
    Solution

    To reflect the graph over the y-axis, replace \( x \) with \( -x \), and to stretch it vertically by a factor of 3, multiply the function by 3. The transformed function is:

    \[ f(x) = 3|x| \]

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