2. Graphical Transformations of Functions
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1 2. Graphical Transformations of Functions In this section we will discuss how the graph of a function may be transformed either by shifting, stretching or compressing, or reflection. In this section let c be a positive real number. Vertical Translations A shift may be referred to as a translation. If c is added to the function, where the function becomes, then the graph of will vertically shift upward by c units. If c is subtracted from the function, where the function becomes then the graph of will vertically shift downward by c units. In general, a vertical translation means that every point (, y) on the graph of is transformed to (, y + c) or (, y c) on the graphs of or respectively Horizontal Translations If c is added to the variable of the function, where the function becomes, then the graph of will horizontally shift to the left c units. If c is subtracted from the variable of the function, where the function becomes, then the graph of will horizontally shift to the right c units. In general, a horizontal translation means that every point (, y) on the graph of is transformed to ( c, y) or ( + c, y) on the graphs of or respectively
2 Reflection If the function or the variable of the function is multiplied by -1, the graph of the function will undergo a reflection. When the function is multiplied by -1 where becomes, the graph of is reflected across the - ais. On the other hand, if the variable is multiplied by -1, where, the graph of is reflected across the y-ais. 3 becomes f f 3
3 Vertical Stretching and Shrinking If c is multiplied to the function then the graph of the function will undergo a vertical stretching or compression. So when the function becomes and, a vertical shrinking of the graph of will occur. Graphically, a vertical shrinking pulls the graph of toward the -ais. When in the function, a vertical stretching of the graph of will occur. A vertical stretching pushes the graph of away from the -ais. In general, a vertical stretching or shrinking means that every point (, y) on the graph of is transformed to (, cy) on the graph of Horizontal Stretching and Shrinking If c is multiplied to the variable of the function then the graph of the function will undergo a horizontal stretching or compression. So when the function becomes and, a horizontal stretching of the graph of will occur. Graphically, a vertical stretching pulls the graph of away from the y-ais. When in the function, a horizontal shrinking of the graph of will occur. A horizontal shrinking pushes the graph of toward the y-ais. In general, a horizontal stretching or shrinking means that every point (, y) on the graph of is transformed to (/c, y) on the graph of. f
4 Transformations can be combined within the same function so that one graph can be shifted, stretched, and reflected. If a function contains more than one transformation it may be graphed using the following procedure: Steps for Multiple Transformations Use the following order to graph a function involving more than one transformation: 1. Horizontal Translation 2. Stretching or shrinking 3. Reflecting 4. Vertical Translation Eamples: Graph the following functions and state their domain and range: basic function ( b.f.) = 2, 2, , 3 b.f. = 2 Domain = Range = [ 3
5 2. 3 b.f. =, 3, reflect about -ais, 1 b.f. = 3 3 reflect about -ais Domain = Range =
6 b.f. =, 3, stretch about y-ais (c = 2), reflect about -ais, 1 b.f. = 3 3 Stretch about y-ais 2 3 Reflect about -ais Domain =[ 3 Range =
7 b.f. = 3, 1, shrink about -ais (c = 2), reflect about y-ais, 2 b.f. = Shrink about -ais Reflect about y-ais Domain = Range =
8 5. Let the graph of f() be the following: Graph the following problems: a b. 1 c. 2 2, 1 2
9 d. 1 Reflect about -ais e. 2 Reflect about y-ais f 2 f 2
10 Transformations of the graphs of functions shift up c units shift down c units shift left c units shift right c units reflect about the y-ais reflect about the -ais When vertical shrinking of When vertical stretching of Multiply the y values by c When horizontal stretching of When horizontal shrinking of Divide the values by c
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