MAT 106: Trigonometry Brief Summary of Function Transformations

Transcription

1 MAT 106: Trigonometry Brief Summary of Function Transformations The sections below are intended to provide a brief overview and summary of the various types of basic function transformations covered in this course. Detailed explanations are not included, but specific examples are given based on the following parent functions: f(x) = x g(x) = x 2 h(x) = x

2 Vertical Translation ("Slide" or "Shift") A vertical translation moves a graph up or down by adding to or subtracting from, respectively, the parent function. Graphically speaking, all x-values remain unchanged, but all y-values are modified according to the specific value used to accomplish the vertical translation. Example 1: Move f(x) up 7 units. f(x) = x + 7 Example 2: Move g(x) down 3 units. g(x) = x 2 3 Example 3: Move h(x) down 4 units. h(x) = x 4 Horizontal Translation ("Slide" or "Shift") A horizontal translation moves a graph left or right by subtracting from or adding to, respectively, the independent variable in the parent function. Graphically speaking, all y-values remain unchanged, but all x-values are modified according to the specific value used to accomplish the horizontal translation. Example 1: Move f(x) left 8 units. f(x) = x + 8 Example 2: Move g(x) right 25 units. g(x) = (x 25) 2 Example 3: Move h(x) left 17 units. h(x) = x + 17

3 Vertical Scale Change ("Stretch" or "Compress") A vertical scale change vertically stretches or compresses by multiplying or dividing, respectively, the parent function by a positive value greater than 1. Alternatively, a vertical stretch can be accomplished by dividing by a positive value less than 1, and a vertical compression can be accomplished by multiplying by a positive value less than 1. Note that multiplying or dividing by 1 causes no change in the graph, and multiplying or dividing by a negative value causes a reflection, which is discussed later. Also, note that multiplying by 0 collapses any function into the horizontal axis, and dividing by 0 is a mathematical impossibility. Graphically speaking, all x-values remain unchanged, but all y-values are modified according to the specific value used to accomplish the vertical scale change. Example 1: Compress f(x) vertically by a factor of 5. f(x) = x 5 = 1 5 x Example 2: Stretch g(x) vertically by a factor of 2. g(x) = 2x 2 Example 3: Stretch h(x) vertically by a factor of 9. h(x) = 9 x Horizontal Scale Change ("Stretch" or "Compress") A horizontal scale change horizontally stretches or compresses by dividing or multiplying, respectively, the independent variable in the parent function by a positive value greater than 1. Alternatively, a horizontal stretch can be accomplished by multiplying by a positive value less than 1, and a horizontal compression can be accomplished by dividing by a positive value less than 1. Note that multiplying or dividing by 1 causes no change in the graph, and multiplying or dividing by a negative value causes a reflection, which is discussed later. Also, note that multiplying by 0 collapses any function into the vertical axis, and dividing by 0 is a mathematical impossibility. Graphically speaking, all y-values remain unchanged, but all x-values are modified according to the specific value used to accomplish the horizontal scale change. Example 1: Compress f(x) horizontally by a factor of 6. f(x) = 6x Example 2: Compress g(x) horizontally by a factor of 15. g(x) = (15x) 2 Example 3: Stretch h(x) horizontally by a factor of 10. h(x) = x 10 = 1 10 x

4 Reflection Across (Over, In, Through, or Off) the Horizontal (x) Axis A reflection across the horizontal axis results in a mirror image using the horizontal axis as a reflecting line by negating (multiplying by 1) the parent function. Graphically speaking, all x-values remain unchanged, but all y-values become their opposites. Example 1: Reflect f(x) across the x-axis. f(x) = x Example 2: Reflect g(x) across the x-axis. g(x) = x 2 Example 3: Reflect h(x) across the x-axis. h(x) = x Reflection Across (Over, In, Through, or Off) the Vertical (y) Axis A reflection across the vertical axis results in a mirror image using the vertical axis as a reflecting line by negating (multiplying by 1) the independent variable in the parent function. Graphically speaking, all y-values remain unchanged, but all x-values become their opposites. Example 1: Reflect f(x) across the y-axis. f(x) = x Example 2: Reflect g(x) across the y-axis. g(x) = ( x) 2 Example 3: Reflect h(x) across the y-axis. h(x) = x

5 Combinations of Function Transformations The various preceding function transformations can be combined such that more than one of them is applied to a single function simultaneously. To apply multiple transformations, the order of operations is applied such that all vertically-based transformations follow the standard order, but all horizontallybased transformations are applied in reverse order. Due to the use of explicit or implied grouping symbols, all horizontally-based transformations are applied before any vertically-based transformations. Example 1: Move f(x) 4 units left, then move it 5 units down. f(x) = (x + 4) 5 Example 2: Move g(x) 7 units right, then reflect it over the x-axis. g(x) = (x 7) 2 Example 3: Compress h(x) horizontally by a factor of 2, then reflect it over the y-axis. h(x) = 2x Example 4: Move f(x) 3 units right, then reflect it over the y-axis, then stretch it vertically by a factor of 6. f(x) = 6( x 3) Example 5: Reflect g(x) over the y-axis, then reflect it over the x-axis, then compress it vertically by a factor of 4, then move it 8 units up. g(x) = 1 4 ( x)2 + 8 = ( x) Example 6: Move h(x) 9 units right, then compress it horizontally by a factor of 4, then reflect it over the x-axis, then stretch it vertically by a factor of 2, then move it 7 units up. h(x) = 2 4x Comments: Vertically-based transformations (those that pertain to the y-values) flow logically in that moving up involves addition, moving down involves subtraction, stretching involves multiplication, and compressing involves division, and these transformations are applied in sequence following the standard order of operations. However, horizontally-based transformations (those that pertain to the x-values) reverse such logic in that moving right involves subtraction, moving left involves addition, stretching involves division, and compressing involves multiplication, and these transformations are performed in sequence following the order of operations in reverse order.

6 Illustrations of Function Transformations The images on the following pages illustrate the results of applying the various transformations discussed above using the specific examples on the preceding pages. Each graph shows the appropriate parent function along with the function obtained after applying the necessary transformation(s). The illustrations follow the order of the examples as presented on the preceding pages and include legends for reference.

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