Dilations can be broken into two types: a compression when the dilation is a proportional decrease, and a stretch when the dilation is a proportional increase.

Each of these can be further categorized as horizontal (stretching away from or compressing towards the y-axis) or vertical (stretching away from or compressing towards the x-axis).

Vertical dilations can be represented algebraically byg\left(x\right) = af\left(x\right)where 0 < a < 1 corresponds to a compression and a > 1 corresponds to a stretch.

Similarly, horizontal dilations can be represented by g\left(x\right) = f\left(\dfrac{x}{a}\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

As with dilations, 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.

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

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Vertical compression with scale factor of 0.25: \\g\left(x\right) = 0.25f\left(x\right)

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

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

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

Worked examples

Example 1

Consider the quadratic function y = \left(x - 2\right)^2

x	-2	-1	0	1	2	3
y

Complete the table of values

Approach

We want to evaluate y at each value of x in the table by substituting into y = \left(x - 2\right)^2; that is, for each x-value, subtract 2 and square the result to get the corresponding y-value.

Solution

x	-2	-1	0	1	2	3
y	16	9	4	1	0	1

Sketch a graph of the parabola

Approach

We can plot points from the table of values and then draw the curve connecting them.

Solution

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-1

Reflection

Note that we can also see certain features of the graph just by looking at the table of values.

Remember that the x-intercepts occur when y = 0, and the y-intercept occurs when x = 0.

Describe the transformation of the graph of y = x^2 that results in the graph of y = \left(x - 2\right)^2

Approach

It will be easiest to see the transformation that has occurred by adding the graph of y = x^2 to the same coordinate plane as the previous graph:

Solution

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The graph of y = x^2 has been added as a dashed line. By looking at corresponding points on the two graphs, we can see that the graph of y = \left(x - 2\right)^2 is a translation of 2 units to the right.

Example 2

Point A\left(-3, 7\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) = f\left(x + 4\right) + 1

Approach

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.

Solution

In the expression f\left(x + 4\right) + 1, the +4 inside of function f indicates a horizontal translation of 4 units to the left, while the +1 outside of function f indicates a vertical translation of 1 unit upwards.

Applying these translations to point A\left(-3, 7\right) results in the point \left(-7, 8\right).

Reflection

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) = 7. We can rewrite this to be in the right form for g\left(x\right) as follows:

\displaystyle f\left(-3\right)	\displaystyle =	\displaystyle 7	Known point
\displaystyle f\left(-7 + 4\right)	\displaystyle =	\displaystyle 7	Rewrite -3 in the form ⬚ + 4
\displaystyle f\left(-7 + 4\right) + 1	\displaystyle =	\displaystyle 8	Add 1 to both sides
\displaystyle g\left(-7\right)	\displaystyle =	\displaystyle 8	Use definition of g\left(x\right)

So the corresponding point is \left(-7, 8\right).

Example 3

In the table of values below, function g has been obtained by transforming function f. Determine an expression for g in terms of f.

x	f\left(x\right)	g\left(x\right)
2	6	3
3	8	4
4	6	3
5	4	2
6	2	1

Approach

We want to look for any patterns between the values in the columns for f and g.

Solution

Notice that f has a maximum value (of 8) when x = 3, and g also takes its maximum value (of 4) at this point. Since both functions have a maximum at this point, there hasn't been a horizontal translation or any reflections.

In this case, each value for f is twice as big as the corresponding value for g - that is,\\f\left(x\right) = 2g\left(x\right) is true for all values in the table. Rewriting this, we get g\left(x\right) = \frac{1}{2}f\left(x\right)

Reflection

We can also plot the points in the table to confirm our answer (or to help find the relationship in the first place):

Example 4

A function is shown in the graph below. Determine an equation for the function after it has been reflected across the x-axis and translated 4 units to the left.

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Approach

There are two main approaches we can use here:

Apply the transformations to the graph, then determine the equation of the final graph
Determine the equation of the graph shown, then apply the transformations algebraically

Solution

Choosing to apply the transformations to the graph first results in the following:

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Looking at the final graph, we can see that it is a straight line which passes through the origin and has a slope of -\dfrac{1}{2}. So an equation for this function is y = -\frac{x}{2}

Reflection

The original function has an equation of y = \dfrac{x}{2} - 2.

A reflection across the x-axis is represented by changing the signs of all the function values (i.e. multiplying throughout by -1), so this transformation results in y = -\dfrac{x}{2} + 2.

A horizontal translation of 4 units to the left is represented by adding 4 directly to the x-values. So this transformation results in

\displaystyle y	\displaystyle =	\displaystyle -\frac{x + 4}{2} + 2
	\displaystyle =	\displaystyle -\frac{x}{2} - \frac{4}{2} + 2
	\displaystyle =	\displaystyle -\frac{x}{2} - 2 + 2
	\displaystyle =	\displaystyle -\frac{x}{2}

which is the same result.

Outcomes

A2.F.BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

A2.MP8

Look for and express regularity in repeated reasoning.

1.02 Transformations of functions

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Reflection

Approach

Solution

Example 2

Approach

Solution

Reflection

Example 3

Approach

Solution

Reflection

Example 4

Approach

Solution

Reflection

Outcomes

A2.F.BF.B.3

A2.MP1

A2.MP2

A2.MP3

A2.MP6

A2.MP7

A2.MP8

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