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Parent Graphs and Transformations A picture of a graph can easily be created by using transformations on the parent graph of a function. This is helpful in saving time by not having to do needless calculations. The following concept can be applied. Shift of a Graph: When the function of the graph is y = ±f(x – h) + k, then a shift of the original graph of y = f(x) can be created by moving h-units in the x-direction and k-units in the y-direction. If there is an negative sign in front of the function, then the graph is reflected over the x-axis. EX.1 f(x) = |x – 3| + 5 10 y = |x-3| + 5 8 6 4 2 y = |x| 5 units up 0 x 3 units right -2 -4 -6 -8 -10 -5 -4 -3 -2 -1 0 y 1 2 3 4 5 EX.2 f(x) = -(x + 4)2 – 3 15 10 5 f(x) = x2 0 x 3 units down 4 units left -5 -10 f(x) = -(x+4)2 - 3 -15 -10 The Math Center -8 ■ -6 -4 Valle Verde -2 0 y ■ 2 4 6 Tutorial Support Services 8 ■ 10 EPCC Constant Function (y = c) Linear Function ( y = x ) y 5 25 4 4 20 3 3 15 2 2 10 1 1 5 5 0 x -1 -1 -5 -2 -2 -10 -3 -3 -15 -4 -4 -20 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 y 1 2 3 4 -25 -5 5 -4 -3 Square Root Function ( y = x1/2 ) Absolute Value Function ( y = |x| ) -2 -1 0 1 2 3 4 5 Cube Function ( y = x3 ) 5 5 4 4 100 3 3 50 2 2 1 1 x 0 x 0 x 0 -1 -1 -50 -2 -2 -3 -3 -100 -4 -4 -5 -5 -5 -5 x 0 x 0 Square Function (y = x2) y -4 -3 -2 -1 0 y 1 2 3 4 5 -5 -5 -4 -3 -2 -1 0 y 1 2 3 4 -5 5 -4 -3 150 5 -1 0 y 1 2 3 4 5 Cube Root Function (y = x1/3) Exponential Function (y = ax) Logarithmic Function (y = log x) -2 5 4 4 100 3 3 2 2 50 1 1 0 x 0 x 0 x -1 -1 -50 -2 -2 -3 The Math Center -100 ■ -3 Valle Verde ■ Tutorial Support Services ■ -4 -4 -5 -5 EPCC -4 -3 -2 -1 0 y 1 2 3 4 5 -150 -5 -4 -3 -2 -1 0 y 1 2 3 4 5 -5 -5 -4 -3 -2 -1 0 y 1 2 3 4 5