IntroductionTransformations can be made to functions in two distinct ways: by transforming the core variable of the function (multiplying the independent variable by k), and by transforming the function as a whole (multiplying the dependent variable by k). Previously, we saw how adding some constant k to the variable of a function or to the entire function affected the graph of the function.

In this lesson, we will see how multiplying by a constant affects the graph of a function. Given f(x) and a constant k, we will observe the transformations f(k • x) and k • f(x).

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key ConceptsGraphing and Points of Interest • In the graph of a function, there are key points of

interest that define the graph and represent the characteristics of the function.

• When a function is transformed, the key points of the graph define the transformation.

• The key points in the graph of a quadratic equation are the vertex and the roots, or x-intercepts.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continuedMultiplying the Dependent Variable by a Constant, k: k • f(x) • In general, multiplying a function by a constant will

stretch or shrink (compress) the graph of f vertically. • If k > 1, the graph of f(x) will stretch vertically by a

factor of k (so the parabola will appear narrower). • A vertical stretch pulls the parabola and stretches it

away from the x-axis. • If 0 < k < 1, the graph of f(x) will shrink or compress

vertically by a factor of k (so the parabola will appear wider).

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued• A vertical compression squeezes the parabola

toward the x-axis. • If k < 0, the parabola will be first stretched or

compressed and then reflected over the x-axis. • The x-intercepts (roots) will remain the same, as will

the x-coordinate of the vertex (the axis of symmetry). • While k • f(x) = f(k • x) can be true, generally

k • f(x) ≠ f(k • x).

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Vertical stretches: when k > 1 in k • f(x)

• The graph is stretched vertically by a factor of k.

• The x-coordinate of the vertex remains the same.

• The y-coordinate of the vertex changes.

• The x-intercepts remain the same.

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Vertical compressions: when 0 < k < 1 in k • f(x)

• The graph is compressed vertically by a factor of k.

• The x-coordinate of the vertex remains the same.

• The y-coordinate of the vertex changes.

• The x-intercepts remain the same.

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Reflections over the x-axis: when k = –1 in k • f(x)

• The parabola is reflected over the x-axis.

• The x-coordinate of the vertex remains the same.

• The y-coordinate of the vertex changes.

• The x-intercepts remain the same.

• When k < 0, first perform the vertical stretch or compression, and then reflect the function over the x-axis.

Key Concepts, continuedMultiplying the Independent Variable by a Constant, k: f(k • x) • In general, multiplying the independent variable in a

function by a constant will stretch or shrink the graph of f horizontally.

• If k > 1, the graph of f(x) will shrink or compress

horizontally by a factor of (so the parabola will

appear narrower). • A horizontal compression squeezes the parabola

toward the y-axis.8

5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued

• If 0 < k < 1, the graph of f(x) will stretch horizontally by

a factor of (so the parabola will appear wider).

• A horizontal stretch pulls the parabola and stretches it away from the y-axis.

• If k < 0, the graph is first horizontally stretched or compressed and then reflected over the y-axis.

• The y-intercept remains the same, as does the y-coordinate of the vertex.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued• When a constant k is multiplied by the variable x of a

function f(x), the interval of the intercepts of the function is increased or decreased depending on the value of k.

• The roots of the equation ax2 + bx + c = 0 are given by

the quadratic formula,

• Remember that in the standard form of an equation, ax2 + bx + c, the only variable is x; a, b, and c represent constants.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued• If we were to multiply x in the equation ax2 + bx + c by

a constant k, we would arrive at the following:

• Use the quadratic formula to find the roots of

, as shown on the following slide.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Quadratic equation

Substitute ak2 for a and bk for b based on the equation in standard form.

Simplify.

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Horizontal compressions: when k > 1 in f(k • x)

• The graph is

compressed horizontally

by a factor of

• The y-coordinate of the vertex remains the same.

• The x-coordinate of the vertex changes.

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Horizontal stretches: when 0 < k < 1 in f(k • x)

• The graph is stretched

horizontally by a factor of

• The y-coordinate of the vertex remains the same.

• The x-coordinate of the vertex changes.

Key Concepts, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Reflections over the y-axis: when k = –1 in f(k • x)

• The parabola is reflected over the y-axis.

• The y-coordinate of the vertex remains the same.

• The x-coordinate of the vertex changes.

• The y-intercept remains the same.

• When k < 0, first perform the horizontal compression or stretch, and then reflect the function over the y-axis.

Common Errors/Misconceptions• thinking that multiplying the dependent variable by a

constant k yields the same equation as multiplying the independent variable by the same constant k in all cases (i.e., that k • f(x) is always equal to f(k • x), but this is not always true)

• forgetting to substitute all values of x with k • x when working with the transformation f(k • x)

• forgetting to square the constant k when substituting f(k • x) into ax2 + bx + c

• confusing horizontal with vertical transformations and vice versa

• confusing stretches and compressions16

5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice

Example 1Consider the function f(x) = x2, its graph, and the constant k = 2. What is k • f(x)? How are the graphs of f(x) and k • f(x) different? How are they the same?

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 1, continued

1. Substitute the value of k into the function. If f(x) = x2 and k = 2, then k • f(x) = 2 • f(x) = 2x2.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 1, continued

2. Use a table of values to graph the functions.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

x f(x) k • f(x)

–2 4 8

–1 1 2

0 0 0

1 1 2

2 4 8

Guided Practice: Example 1, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 1, continued

3. Compare the graphs. Notice the position of the vertex has not changed in the transformation of f(x). Therefore, both equations have same root, x = 0. However, notice the inner graph, 2x2, is more narrow than x2 because the value of 2 • f(x) is increasing twice as fast as the value off(x). Since k > 1, the graph of f(x) will stretch vertically by a factor of 2. The parabola appears narrower.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

✔

Guided Practice: Example 1, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice

Example 2Consider the function f(x) = x2 – 81, its graph, and the constant k = 3. What is f(k • x)? How do the vertices and the zeros of f(x) and f(k • x) compare?

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 2, continued

1. Substitute the value of k into the function. If f(x) = x2 – 81 and k = 3, then f(k • x) = f(3x) = (3x)2 – 81 = 9x2 – 81.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 2, continued

2. Use the zeros and the vertex of f(x) to graph the function.To find the zeros of f(x), set the function equal to 0 and factor.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 2, continued

The zeros are –9 and 9.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

x2 – 81 = 0Set the function equal to 0.

(x + 9)(x – 9) = 0Factor using the difference of two squares.

x + 9 = 0 or x – 9 = 0Use the Zero Product Property.

x = –9 or x = 9 Solve for x.

Guided Practice: Example 2, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

The vertex of f(x) is (0, –81). This can be seen as the translation of the parent function f(x) = x2. The parent function is translated down 81 units.

Guided Practice: Example 2, continued

3. Using the zeros and the vertex of the transformed function, graph the new function on the same coordinate plane.

Set the transformed function equal to 0 and factor to find the zeros.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 2, continued

The zeros are –3 and 3.29

5.8.2: Replacing f(x) with k • f(x) and f(k • x)

9x2 – 81 = 0 Set the function equal to 0.

9(x2 – 9) = 0 Use the GCF to factor out 9.

9(x + 3)(x – 3) = 0 Use the difference of two squares.

x + 3 = 0 or x – 3 = 0 Use the Zero Product Property.

x = –3 or x = 3 Solve for x.

Guided Practice: Example 2, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

The vertex is the same as the original function, (0, –81). This again can be seen as the translation of 9x2 down 81 units.

Guided Practice: Example 2, continued

4. Compare the graphs.

Notice the position of the vertex has not changed in

the transformation of f(x). However, notice the inner

graph, f(3x) = 9x2 – 81, is narrower than f(x).

Specifically, the roots are x = –9 and 9 for f(x) and

x = –3 and 3 for f(3x). This is because the roots of the

function f(k • x) are times the roots of f(x) in a

quadratic equation. 31

5.8.2: Replacing f(x) with k • f(x) and f(k • x)

Guided Practice: Example 2, continued

Since k > 1, the graph of f(x) will shrink horizontally by

a factor of , so the parabola appears narrower.

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)

✔

Guided Practice: Example 2, continued

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5.8.2: Replacing f(x) with k • f(x) and f(k • x)