Topics
1. Analyzing Functions
topic badge
United States of AmericaVirginia
Algebra 2

1.03 Function transformations

Lesson
Practice

Function transformations

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.

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)

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Reflection across the x-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
Reflection across the y-axis: {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 \gt 0 translates upwards and k \lt 0 translates downwards.

-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
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.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Horizontal translation of 3 units left: {g\left(x\right) = f\left(x+3\right)}
-3
-2
-1
1
2
3
4
5
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)}
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

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.

-4
-3
-2
-1
1
2
3
4
x
-1.5
-1
-0.5
0.5
1
1.5
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
-3
-2
-1
1
2
3
y
Vertical stretch with a scale factor of 2: {g\left(x\right) = 2f\left(x\right)}
Horizontal compression

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

Horizontal stretch

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

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}}

-1
1
x
1
2
3
y
Horizontal compression with a scale factor of 0.5: {g\left(x\right)=f\left(2x\right)}
-2
-1
1
2
x
1
2
3
y
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.

Examples

Example 1

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).

-4
-3
-2
-1
1
2
3
4
x
-2
-1
1
2
3
4
5
6
y
Worked Solution
Create a strategy

We can first note the transformations that have been applied to the parent function. We can see that g\left(x\right) has been reflected across the y-axis and translated 1 unit left. In function notation, this means {b=-1} and h=-1 which corresponds to g\left(x\right)=f\left[-\left(x + 1\right)\right].

Apply the idea

Because f\left(x\right)=\sqrt{x}, we will substitute -\left(x+1\right) for x underneath the radical.

\displaystyle g\left(x\right)\displaystyle =\displaystyle f\left[-\left(x + 1\right)\right]
\displaystyle g\left(x\right)\displaystyle =\displaystyle \sqrt{-\left(x+1\right)}

The equation of the transformed function is g\left(x\right)=\sqrt{-\left(x+1\right)} or g\left(x\right)=\sqrt{-x-1}.

Reflect and check

When identifying transformations that have been applied to a function, it is important to identify them in the correct order. We should always look for reflections and dilations first, then translations.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

If we had written the translation first and the reflection second, we would have mistakely found the equation to be g\left(x\right)=\sqrt{-x+1}.

As shown in the graph, this is not the same function as the given one. Additionally, notice that the coefficient of x is factored out when written in function notation {g\left(x\right)=f\left[b\left(x-h\right)\right]} That helps us see the function is equivalent to g\left(x\right)=\sqrt{-\left(x-1\right)} which represents a reflection across the y-axis and a translation to the right 1 unit.

Example 2

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
Create a strategy

We can use the given expression to determine the transformations from f\left(x\right) to g\left(x\right), then apply these transformations to the point A.

Apply the idea

In the expression \dfrac{1}{3}\cdot f\left(x + 4\right), the +4 inside of function f indicates a horizontal translation of 4 units to the left, while the \dfrac{1}{3} outside of function f indicates a vertical shrink by a factor of \dfrac{1}{3}.

Shifting the point left 4 units takes the point to \left(-7, 9\right), and shrinking it vertically by 3 requires multiplying the y-coordinate by \dfrac{1}{3}.

9 \cdot \dfrac{1}{3}=3 gives us the point \left(-7, 3\right).

Reflect and check

We can also think about these transformations algebraically.

The only point we know on the graph of f is A, which tells us that f\left(-3\right) = 9. We can rewrite this to be in the correct form for g\left(x\right) as follows:

\displaystyle f\left(-3\right)\displaystyle =\displaystyle 9Known point
\displaystyle f\left(-7 + 4\right)\displaystyle =\displaystyle 9Rewrite -3 in the form ⬚ + 4
\displaystyle \dfrac{1}{3}\cdot f\left(-7 + 4\right)\displaystyle =\displaystyle 3Multiply both sides by \dfrac{1}{3}
\displaystyle g\left(-7\right)\displaystyle =\displaystyle 3Use definition of g\left(x\right)

So, the corresponding point is \left(-7, 3\right).

Example 3

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

-2
-1
1
2
x
-1
1
2
3
4
f\left(x\right)
a

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

Worked Solution
Create a strategy

The graph is a parabola, which tells us that it is a quadratic function. The parent quadratic function is f(x)=x^{2}. We need to give the transformed function a new name, like g\left(x\right), since we are creating a new function.

A horizontal translation and a horziontal stretch are represented by b and h in the function notation g\left(x\right)=f\left[b\left(x-h\right)\right]. Remember b=\dfrac{1}{\text{stretch factor}} for horizontal stretches and compressions.

Apply the idea

In this case, the stretch factor is 2, so b=\dfrac{1}{2}. To translate the function 6 units right, h=6. To find the equation after it has been horizontally translated and stretched, we need to find g\left(x\right)=f\left[\dfrac{1}{2}\left(x-6\right)\right].

\displaystyle g\left(x\right)\displaystyle =\displaystyle f\left[\dfrac{1}{2}\left(x-6\right)\right]
\displaystyle =\displaystyle \dfrac{1}{2}(x-6)^{2}

The equation of the transformed function is g\left(x\right)=\dfrac{1}{2}(x-6)^{2}, or g\left(x\right)=\dfrac{1}{2}x^{2}-6x+18 in standard form.

Reflect and check

If we expand the polynomial into standard form, the transformation becomes difficult to determine. Looking at the standard form of the quadratic would not tell us the transformations that took place.

In order to identify the transformations when given an equation, we must factor the polynomial completely until it resembles the standard function notation:g\left(x\right)=a\cdot f\left[b\left(x-h\right)\right]+k

By first factoring out a greatest common factor, then factoring the remaining trinomial, we see that g(x)=\dfrac{1}{2}(x^{2}-12x+36)

g(x)= \dfrac{1}{2}(x-6)^{2} which shows that the parent quadratic function was translated 6 units to the right and stretched horizontally by a factor of 2.

b

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

Worked Solution
Create a strategy

A horizontal stretch will make the graph wider, and horizontal translation of 6 units to the right will move the vertex to \left(6,0\right). We will stretch the graph horizontally to determine the correct shape of the graph, then we will shift it to the right.

Apply the idea

Using a table of values for f\left(x\right) can help us stretch the function horizontally. A table of values for the parent quadratic function is:

x-3-2-10123
f\left(x\right)9410149

Since the graph shifts to the right, we will shift each input value by 6 and evaluate to determine the new output values.

x3456789
y4.520.500.524.5

Using the tables to graph both functions, we can see we have stretched f\left(x\right) horizontally by 2.

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

Now, we simply need to shift all the points on the stretched graph to the right 6 units.

-2
-1
1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
y
Reflect and check

In general, when the transformations are both horizontal transformations (or a reflection across the y-axis), apply the reflections, stretches, and compressions first. Apply the translations last. The same rule applies when the transformations are both vertical (or a reflection across the x-axis).

Example 4

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
Create a strategy

When we compare f\left(x\right) and g\left(x\right), we can see that there are numbers that have been added and subtracted. Addition and subtraction represent translations.

Apply the idea

5 is subtracted from the input values and 6 is substracted from the output values. In function notation, it can be represented as:

g\left(x\right)=f\left(x-5\right)-6

The graph of g\left(x\right) is the graph of f\left(x\right) after it has been translated right 5 units and down 6 units.

Reflect and check

When we graph the functions, we can see that f\left(x\right) has been shifted right 5 units and down 6 units to obtain the graph of g\left(x\right).

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Idea summary

The reflections and translations can be summarized as follows:

A figure showing a summary of reflections and translation. The expression a f of left bracket b left parenthesis x minus h right parenthesis right bracket plus k is shown. Under the column labeled vertical: for a: when a is greater than 0, no reflection; when a is less than 0, reflection across the x-axis; when absolute value of a is greater than 1, vertical stretch; and when absolute value of a is between 0 and 1, vertical compression; for k: when k is greater than 0, vertical translation up; and when k is less than 0, vertical translation down. Under the column labeled horizontal: for b: when b is greater than 0, no reflection; when b is less than 0, reflection across the y-axis; when absolute value of b is greater than 1, horizontal compression; and when absolute value of b is between 0 and 1, horizontal stretch; for h: when h is greater than 0, horizontal translation right; and when h is less than 0, horizontal translation left.

Outcomes

A2.F.1

The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.F.1b

Write the equation of a square root, cube root, rational, exponential, and logarithmic function, given a graph, using transformations of the parent function, including f(x) + k; f(kx); f(x + k); and kf(x), where k is limited to rational values. Transformations of exponential and logarithmic functions, given a graph, should be limited to a single transformation.

What is Mathspace

About Mathspace