Topics
1. Analyzing Functions
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United States of AmericaFL
Grade 912

1.02 Transformations of functions

Lesson
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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:

Dilation

A proportional increase or decrease in size in all directions

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-2-10123
y
a

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-2-10123
y1694101
b

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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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.

c

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

xf\left(x\right)g\left(x\right)
263
384
463
542
621

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):

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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:

-4
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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

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

MA.912.F.2.2

Identify the effect on the graph of a given function of two or more transformations defined by adding a real number to the x- or y- values or multiplying the x- or y- values by a real number.

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.

MA.912.F.2.5

Given a table, equation or graph that represents a function, create a corresponding table, equation or graph of the transformed function defined by adding a real number to the x- or y-values or multiplying the x- or y-values by a real number.

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