A **transformation** of a function is a change in the position or shape of its graph. The function in any family with the simplest form is known as the **parent function**. A **function family** consists of the parent function and any function that can be created by applying a series of transformations to the parent function.

We frequently consider transformations as coming from the parent function. In the examples of transformations that follow, the parent function is shown as a dashed line.

A reflection across the x-axis can be represented algebraically by g\left(x\right) = -f\left(x\right)A reflection across the y-axis can be represented algebraically by g\left(x\right) = f\left(-x\right)

Reflection across the x-axis: {g\left(x\right)=-f\left(x\right)}

Reflection across the y-axis: {g\left(x\right)=f\left(-x\right)}

Translations can be categorized as **horizontal** (moving left or right, along the x-axis) or **vertical** (moving up or down, along the y-axis), or a combination of the two.

Vertical translations can be represented algebraically by g\left(x\right) = f\left(x\right) + kwhere k \gt 0 translates upwards and k \lt 0 translates downwards.

Vertical translation of 4 units upwards: {g\left(x\right) = f\left(x\right) + 4}

Vertical translation of 4 units downwards: {g\left(x\right)=f\left(x\right)-4}

Similarly, horizontal translations can be represented by g\left(x\right) = f\left(x - h\right) where h > 0 translates to the right and h < 0 translates to the left.

Horizontal translation of 3 units left: {g\left(x\right) = f\left(x+3\right)}

Horizontal translation of 3 units to the right: {g\left(x\right) = f\left(x - 3\right)}

Compressions and stretches are more generally called **dilations**.

Vertical dilations can be represented algebraically by g\left(x\right) = af\left(x\right)where 0 \lt \left|a\right| \lt 1 corresponds to a compression and \left|a\right| \gt 1 corresponds to a stretch.

Vertical compression with scale factor of 0.5: {g\left(x\right) = 0.5f\left(x\right)}

Vertical stretch with a scale factor of 2: {g\left(x\right) = 2f\left(x\right)}

Horizontal dilations can be represented algebraically by g\left(x\right) = f\left(bx\right)where \left|b\right| \gt 1 corresponds to a compression and 0 \lt \left|b\right| \lt 1 corresponds to a stretch.

For horizontal stretches and compressions, b=\dfrac{1}{\text{scale factor}}

Horizontal compression with a scale factor of 0.5: {g\left(x\right)=f\left(2x\right)}

Horizontal stretch by a scale factor of 2: {g\left(x\right)=f\left(0.5x\right)}

When performing multiple transformations at once, we use the standard function notation a\cdot f\left[b\left(x-h\right)\right]+k with the correct values of a,\,b,\,h and k to apply transformations to f\left(x\right). When given a transformed function, we must convert it back to standard notation to correctly identify the transformations applied to the parent function.

We can use the relationship between an equation and its transformations to write equations and sketch graphs.

Consider the graph of the parent function f\left(x\right) and the transformed function g\left(x\right). Write the equation of g\left(x\right).

Worked Solution

Point A\left(-3, 9\right) lies on the graph of f\left(x\right). Determine the coordinates of the corresponding point on the graph of g\left(x\right) = \dfrac{1}{3}\cdot f\left(x + 4\right).

Worked Solution

The graph of a function f\left(x\right) is shown below.

a

Determine the equation after the function has been translated 6 units right and horizontally stretched by a factor of 2.

Worked Solution

b

Graph g\left(x\right) and f\left(x\right) on the same coordinate plane.

Worked Solution

Describe how g\left(x\right)=3^{x-5}-6 has been transformed from its parent function, f\left(x\right)=3^{x}.

Worked Solution

Idea summary

The reflections and translations can be summarized as follows: