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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)

• 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)

• 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)

• 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)

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

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

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

• 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)

• 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)

• 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)

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

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

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

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)

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) = x2 and k • f(x) = 2 • f(x) = 2x2

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

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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