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. 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 : Replacing f(x) with k f(x) and f(k x)2

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) : Replacing f(x) with k f(x) and f(k x)3

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) : Replacing f(x) with k f(x) and f(k x)4

Vertical Stretches 5.8.2: Replacing f(x) with k f(x) and f(k x)5 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.

Vertical Compressions 5.8.2: Replacing f(x) with k f(x) and f(k x)6 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.

Reflections over the x-axis 5.8.2: Replacing f(x) with k f(x) and f(k x)7 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.

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 : Replacing f(x) with k f(x) and f(k x)8

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 : Replacing f(x) with k f(x) and f(k x)9

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 ax 2 + bx + c = 0 are given by the quadratic formula, Remember that in the standard form of an equation, ax 2 + bx + c, the only variable is x; a, b, and c represent constants : Replacing f(x) with k f(x) and f(k x)10

Key Concepts, continued. If we were to multiply x in the equation ax 2 + 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)11

Horizontal Compressions 5.8.2: Replacing f(x) with k f(x) and f(k x)12 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.

Horizontal Stretches 5.8.2: Replacing f(x) with k f(x) and f(k x)13 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.

Reflections over the y-axis 5.8.2: Replacing f(x) with k f(x) and f(k x)14 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.

Practice # 1 Consider the function f(x) = x 2, 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)15

Substitute the value of k into the function. If f(x) = x 2 and k = 2, then k f(x) = 2 f(x) = 2x 2. Use a table of values to graph the functions : Replacing f(x) with k f(x) and f(k x)16 xf(x)f(x)k f(x) –248 –

Graph f(x) = x 2 and k f(x) = 2 f(x) = 2x : Replacing f(x) with k f(x) and f(k x)17

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, 2x 2, is more narrow than x 2 because 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 : Replacing f(x) with k f(x) and f(k x)18

THANKS FOR WATCHING! Ms. Dambreville