Replacing f x with k f x and f k x
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Replacing f ( x ) with k • f ( x ) and f ( k • x ). Adapted from Walch Education. Graphing 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.

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

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Replacing f x with k f x and f k x

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

Adapted from Walch Education


Graphing and points of interest

Graphing 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.

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


Multiplying the dependent variable by a constant k k f x

Multiplying 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).

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


Key concepts continued

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).

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


Vertical stretches

Vertical Stretches

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


Vertical compressions

Vertical Compressions

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


Reflections over the x axis

Reflections over the x-axis

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


Multiplying the independent variable by a constant k f k x

Multiplying 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.

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


Key concepts continued1

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.

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


Key concepts continued2

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.

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


Key concepts continued3

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

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


Horizontal compressions

Horizontal Compressions

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


Horizontal stretches

Horizontal Stretches

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


Reflections over the y axis

Reflections over the y-axis

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


Practice 1

Practice # 1

Consider 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?

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


Substitute the value of k into the function

Substitute the value of k into the function.

If f(x) = x2 and k = 2, then k • f(x) = 2 • f(x) = 2x2.

Use a table of values to graph the functions.

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


Graph f x x 2 and k f x 2 f x 2 x 2

Graph f(x) = x2 and k • f(x) = 2 • f(x) = 2x2

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


Compare the graphs

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 x2because the value of 2 • f(x) is increasing twice as fast as the value of f(x). Since k > 1, the graph of f(x) will stretch vertically by a factor of 2. The parabola appears narrower.

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


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Ms. Dambreville

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