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September 25, 2013 (1.6) Graphical Transformations 1. Graph y = √x Then graph the following and describe how the graph transformed.: a. y = √x + 4 ___________ b. y = √x - 4 ___________ c. y = √(x+4) ___________ d. y = √(x-4) ___________ What happens to a function: y = f(x) when: a. y = f(x) + c ___________________ b. y = f(x) - c ___________________ c. y = f(x + c) ___________________ d. y = f(x - c) ___________________ September 25, 2013 Graph y = √x Then graph the following and describe how the graph transformed.: a. y = - √x _____________ b. y = √(-x) ____________ What happens to a function y = f(x) when: a. y = - f(x) ___________________ b. y = f(-x) ____________________ September 25, 2013 Graph y = √(4-x2) Then graph the following and describe how the graph transformed.: a. y = 2√(4-x2) _____________ b. y = 0.5√(4-x2) ___________ What happens to a function y = f(x) when y = c f(x) if: a. c > 1 _______________________ b. 0 < c < 1 _____________________ September 25, 2013 Graph y = √(4-x2) Then graph the following and describe how the graph transformed.: a. y = √(4-(2x)2) ____________ b. y = √(4-(0.5x)2) ___________ What happens to a function y = f(x) when y = f(cx) if: a. c > 1: _________________________ b. 0 < c < 1: _______________________ September 25, 2013 1.6 Graphical Transformations functions that map real numbers to real numbers Rigid transformations: size and shape are unchanged (translations, reflections, or any combination of these) Non-rigid transformations: shape distorted (vertical and horizontal stretches and shrinks) Do Worksheet September 25, 2013 Translations: Horizontal: f(x - c) Vertical: translate right c units f(x + c) translate left c units f(x) + c translate up c units f(x) - c translate down c units September 25, 2013 The figure shows a graph of y = x3. Write an equation for y2 and y3. y = x3 y2 = y3 = September 25, 2013 Reflections y Across the x-axis: y = -f(x) X Across the y-axis: y = f(-x) September 25, 2013 Find an equation for the reflection of f(x) = 5x2 +x across each axis. across x-axis: across y-axis: September 25, 2013 Stretches and Shrinks Vertical: y = c f(x) a stretch by a factor of c if c>1 a shrink by a factor of c if c<1 Horizontal: y=f x c a stretch by a factor of c if c>1 a shrink by a factor of c if c<1 September 25, 2013 Find the equation for each of the following if f(x) = x3 - 16x. 1. g(x) is a vertical stretch of f(x) by a factor of 3. 2. h(x) is a horizontal shrink of f(x) by factor of 1/2. September 25, 2013 The graph of y = x2 undergoes the following transformations, in order. Find the equation of the graph that results. * a horizontal shift 2 units to the right * a vertical stretch by a factor of 3 * a vertical translation 5 units up September 25, 2013 Determine the graph of the composite function y = 2f(x+1) - 3 by describing the sequence of transformations on the graph of y = f(x). September 25, 2013 Graphing Absolute Value Compositions Given the graph of y = f(x), y = f(x) reflect the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged. y = f( x ) replace the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the yaxis unchanged. (The result will show even symmetry) Graph f(x) = 5x3 + 2x graph f(x) graph f( x ) Do Worksheet: Exploration 2