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This section explores various functions and their symmetries through graphical representation. It covers key equations, such as y = - (x^2 - 1), y = x^2 - 1, and y = |x^2 - 1|, showcasing the differences and similarities in their graphs. The text also details tests for different types of symmetry, including x-axis, y-axis, origin, and y = x symmetry. Understanding these concepts is essential for analyzing graphs and recognizing patterns in mathematical functions. Examples provided facilitate practical application of the theory.
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Graph: y = -(x2 – 1) Graph: y = x2 – 1
Graph: y = -|x2 – 1| Graph: y = |x2 – 1|
Graph: y = (-x + 2)2 Graph: y = (x + 2)2
Graph: x = y2 Graph: y = x2
Tests for Symmetry: • x-axis symmetry: (x, y) and (x, -y) are both on the graph • Test: -leave x -plug in –y for y -see if the equations are equal • y-axis symmetry: (x, y) and (-x, y) are both on the graph • Test: -leave y -plug in –x for x -see if the equations are equal
…Tests for Symmetry: • y = x symmetry: (x, y) and (y, x) are both on the graph • Test: -switch x andy -see if the equations are equal • origin symmetry: (x, y) and (-x, -y) are both on the graph • Test: -plug in –x for x -plug in –y for y -see if the equations are equal
Ex. 1: Let f(x) = 2x – 3. Sketch each graph: a. y = -f(x)
Ex. 1: Let f(x) = 2x – 3. Sketch each graph: b. y = |f(x)|
Ex. 1: Let f(x) = 2x – 3. Sketch each graph: c. y = f(-x)
