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. 5.8.2: 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). 5.8.2: 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). 5.8.2: Replacing f(x) with k f(x) and f(k x) 4
Vertical Stretches 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. 5.8.2: Replacing f(x) with k f(x) and f(k x) 5
Vertical Compressions 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. 5.8.2: Replacing f(x) with k f(x) and f(k x) 6
Reflections over the x-axis 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. 5.8.2: Replacing f(x) with k f(x) and f(k x) 7
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 the parabola will appear narrower). (so A horizontal compression squeezes the parabola toward the y-axis. 1 k 5.8.2: 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 1 horizontally by a factor of (so the k 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) 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, x = -b ± b2-4ac. 2a Remember that in the standard form of an equation, ax 2 + 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) 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: a( kx) 2 + b( kx) + c = ( ak 2 ) x 2 + ( bk ) x + c Use the quadratic formula to find the roots of ( ak 2 ) x 2 + bk ( ) x + c 5.8.2: Replacing f(x) with k f(x) and f(k x) 11
Horizontal Compressions Horizontal compressions: when k > 1 in f(k x) The graph is compressed horizontally 1 by a factor of k. The y-coordinate of the vertex remains the same. The x-coordinate of the vertex changes. 5.8.2: Replacing f(x) with k f(x) and f(k x) 12
Horizontal Stretches Horizontal stretches: when 0 < k < 1 in f(k x) The graph is stretched horizontally by a factor of 1 k. The y-coordinate of the vertex remains the same. The x-coordinate of the vertex changes. 5.8.2: Replacing f(x) with k f(x) and f(k x) 13
Reflections over the y-axis 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. 5.8.2: Replacing f(x) with k f(x) and f(k x) 14
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. x f(x) k f(x) 2 4 8 1 1 2 0 0 0 1 1 2 2 4 8 5.8.2: Replacing f(x) with k f(x) and f(k x) 16
Graph f(x) = x 2 and k f(x) = 2 f(x) = 2x 2 5.8.2: 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. 5.8.2: Replacing f(x) with k f(x) and f(k x) 18
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