Transformations of Functions

🧑Zayn Alazawi 📅June 27, 2020 📊Grade 11 Math

Today, we're going to explore the transformations of functions. Think of these transformations as the tools that allow us to alter the shape, position, and size of a function's graph. By the end of our session, you'll be able to confidently navigate through translations, reflections, stretches, and compressions.

I. Translations: Shifting the Landscape

Imagine a function graph as a piece of art on a canvas. Translations are like picking up that canvas and moving it around.

  • Vertical Translations:

    • To shift the graph upwards, we add a constant \(c\) to the function: \(y=f(x)+c\).
    • To shift it downwards, we subtract \(c\): \(y=f(x)-c\).

  • Horizontal Translations:

    • Shifting to the right involves replacing \(x\) with \((x−c)\): \(y=f(x−c)\).
    • Shifting to the left uses \((x+c)\): \(y=f(x+c)\).

II. Reflections: Flipping the Image

Reflections are like holding a mirror up to our function, changing its orientation.

  • Reflections across the x-axis:

    • Multiply the entire function by -1: \(y=−f(x)\).

  • Reflections across the y-axis:

    • Replace \(x\) with \(−x\): \(y=f(-x)\).

III. Stretches and Compressions: Reshaping the Function

Stretches and compressions are like playing with a rubber band, altering the width or height of our function's graph.

  • Vertical Stretches/Compressions:

    • Stretching vertically by a factor \(k\): \(y=kf(x)\).
    • Compressing vertically by \(k\): \(y=\frac{1}{k}f(x)\).

  • Horizontal Stretches/Compressions:

    • Stretching horizontally by \(k\): \(y=f(kx)\).
    • Compressing horizontally by \(k\): \(y=f(\frac{1}{k}​x)\).
Example: Consider the function \(f(x)=x^2\). We will apply the following transformations step by step:
 

  1. Vertical Translation:

    • Let's shift the graph upward by 2 units. The transformed function is \(g(x)=x^2+2\).

  2. Horizontal Translation:

    • Now, let's shift the graph to the right by 3 units. The transformed function is \(h(x)=(x−3)^2\).

  3. Reflection across the x-axis:

    • Reflect the graph across the x-axis. The transformed function is \(j(x)=−x^2\).

  4. Reflection across the y-axis:

    • Reflect the graph across the y-axis. The transformed function is \(k(x)=x^2\) (no change).

  5. Vertical Stretch:

    • Stretch the graph vertically by a factor of 3. The transformed function is \(m(x)=3x^2\).

  6. Horizontal Compression:

    • Compress the graph horizontally by a factor of 2. The transformed function is \(n(x)=(0.5x)^2\).

Desmos Demo:

Click 'edit graph' to view the individual functions.

✍️Practice

  1. Apply a horizontal compression by a factor of 0.5 and a vertical translation of 3 units downward to the function \(f(x)=x^2\). Write down the transformed function and sketch its graph.

    2. \(g(x)=0.5(x^ 2 )−3\)

  2. Apply a sequence of transformations: a vertical compression by a factor of 0.3, a horizontal translation of 2 units to the left, and a reflection across the y-axis to the function \(f(x)=x^2\). Write down the transformed function and sketch its graph.

    3. \(m(x)=−0.3(x+2)^2\)