# 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. $$g(x)=0.5(x^ 2 )−3$$

3. 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.

4. $$m(x)=−0.3(x+2)^2$$