9-1-2 - 1-2 Transformations of Graphs (learn the graphs of...

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Unformatted text preview: 1-2 Transformations of Graphs (learn the graphs of six “elementary functions”, p. 23) Horizontal shifts reference: f(x) = x2 3 2 1 shift left 2: f(x + 2) = (x + 2)2 y x 3 2 1 shift right 2: f(x - 2) = (x - 2)2 y x 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 5 −3 −2 −1 −4 −1 −2 −3 −4 x 1 2 3 4 5 −3 −2 −1 −4 −1 −2 −3 −4 y 12345 −2 −3 −4 Vertical shifts reference: f(x) = x2 3 2 1 shift up 2: f(x) + 2 = x2 + 2 y x 3 2 1 shift down 2: f(x) - 2 = x2 - 2 y x 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 5 −3 −2 −1 −4 −1 1 2 3 4 5 −3 −2 −1 −4 −1 −2 −3 −4 −2 −3 −4 y x −2 −3 −4 1-2 12345 p. 1 Expansions and Contractions (Omit) Reflections across the X-axis reference: f(x) = x2 reflected in the x-axis: -f(x) = - x2 3 2 1 y 3 2 1 x −4 −3 −2 −1 −1 12345 y x −4 −3 −2 −1 −1 −2 −3 −4 12345 −2 −3 −4 REFLECTIONS ACROSS THE Y-AXIS reference: f(x) = x3 reflected in the y-axis: f(-x) = (-x)3 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 1-2 y x 12345 3 2 1 −4 −3 −2 −1 −1 y x 12345 −2 −3 −4 p. 2 Combinations of transformations to graph: g(x) = -(x - 3)2 + 2 (1) note that basic function is f(x) = x2 (2) find transformations that get from x 2 -(x - 3)2 + 2 x2 (x - 3)2 -(x - 3)2 (x - 3)2 -(x - 3)2 -(x - 3)2 + 2 x2 3 2 1 −4 −3 −2 −1 −1 (x - 3)2 y x 12345 −2 −3 −4 −4 −3 −2 −1 −1 −2 −3 −4 3 2 1 y x −4 −3 −2 −1 −1 12345 −2 −3 −4 -(x - 3)2 3 2 1 Transformation shift right by 3 reflect across x-axis shift up by 2 -(x - 3)2 + 2 y x 12345 3 2 1 −4 −3 −2 −1 −1 y x 12345 −2 −3 −4 When you do problems of this type: • you must state the transformations as shown above • but you do not need to draw the intermediate graphs 1-2 p. 3 Piecewise-Defined Functions Example: f(x) = x = 1+x2 x≤0 x>0 • f is defined using functional notation and a formula • different formula for different parts of the domain • we say that f is piecewise-defined • it "jumps" at x = 0 from 0 to 1 • x = 0 is a point of discontinuity • the "solid dot" at (0,0) is used to indicate clearly that the straight line goes all the way to (0,0) inclusive • that is, that f(0) = 0 • the "open dot" at (0,1) indicates that the curve extends all the way to (0,1), but doesn't include that point • discontinuities are always drawn in this way to indicate the value of the function at the point of discontinuity 1-2 p. 4 ...
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