Topics
4. Functions & Linear Relationships
topic badge
United States of AmericaFL
Grade 912

4.06 Transforming linear functions

Lesson
Practice
Lesson

Concept summary

A transformation of a function is a change in the position, size, or shape of its graph. There are many ways functions can be transformed:

Vertical compression

A transformation that scales all of the y-values of a function by a constant factor towards the x-axis

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor away from the x-axis

A vertical compression or stretch can be represented algebraically by g\left(x\right) = af\left(x\right)where 0 < a < 1 corresponds to a compression and a > 1 corresponds to a stretch.

Reflection

A transformation that produces the mirror image of a figure across a 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)

Translation

A transformation in which every point in a figure is moved in the same direction and by the same distance.

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 > 0 translates upwards and k < 0 translates downwards.

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

Functions that can be obtained by performing one or more of these transformations on each other can be collected into groups or families of functions. The function in any family with the simplest form is known as the parent function, and we frequently consider transformations as coming from the parent function.

The parent function of the linear function family is the function y = x.

Some examples of transformations are shown below. In each example, the parent function is shown as a dashed line:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Vertical compression with scale factor of 0.5: \\g\left(x\right) = 0.5f\left(x\right)
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Reflection across y-axis: \\g\left(x\right) = f\left(-x\right)
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Vertical translation of 4 units upwards: \\g\left(x\right) = f\left(x\right) + 4
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Horizontal translation of 3 units to the right: \\g\left(x\right) = f\left(x - 3\right)

Worked examples

Example 1

A graph of the function f\left(x\right) = \dfrac{1}{4}x + 3 is shown below.

-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
a

Describe the transformation given by g\left(x\right) = -2f\left(x\right)

Solution

The transformation g(x) is a combination of a vertical stretch of the graph of f(x) by a factor of 2, and a reflection across the x-axis.

b

Draw a graph of g\left(x\right) on the same plane as the graph of f\left(x\right).

Approach

To perform a vertical stretch, we want to multiply the y-coordinate of each point by 2 without changing the x-coordinate, so that it is twice as far away from the x-axis.

To perform a reflection across the x-axis, we want to change the sign of the y-coordinate of each point, so that it is on the opposite side of the x-axis.

Solution

-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y

Example 2

The linear functions f\left(x\right) and g\left(x\right) are represented on the given graph.

-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
y
a

Describe the type of transformation(s) that transforms f\left(x\right) to g\left(x\right).

Approach

We can look at the slope and intercepts to help describe which transformation has taken place.

Solution

f\left(x\right) is increasing, while g\left(x\right) is decreasing so there has been a reflection over the x-axis.

g\left(x\right) is steeper than f\left(x\right), so there has also been a dilation.

b

Write an equation for g\left(x\right) in terms of f\left(x\right).

Approach

The equation will be of the form g\left(x\right)=k\cdot f\left(x\right), where k is a real number.

Solution

The slope of f\left(x\right) is \dfrac{1}{2} and the slope of g\left(x\right) is -1, so it has be reflected and is twice as steep.

The transformation is g\left(x\right)=-2 \cdot f\left(x\right).

c

Create a table of values for f\left(x\right) and g\left(x\right) on the same coordinate plane to confirm your answer to parts (a) and (b).

Approach

We can use about five corresponding points to check that g\left(x\right)=-2 \cdot f\left(x\right). Since f\left(x\right) has slope of \dfrac{1}{2} we can use even x-values to avoid using fractions or decimals.

Solution

Getting this data from the graph:

-4
-3
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
3
y
x-4-2024
f\left(x\right)-10123
g\left(x\right)20-2-4-6

This table does confirm that g\left(x\right)=-2 \cdot f\left(x\right).

Outcomes

MA.912.F.2.1

Identify the effect on the graph or table of a given function after replacing f(x) by f(x)+k,kf(x), f(kx) and f(x+k) for specific values of k.

MA.912.F.2.3

Given the graph or table of f(x) and the graph or table of f(x)+k,kf(x), f(kx) and f(x+k), state the type of transformation and find the value of the real number k.

What is Mathspace

About Mathspace