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Graphs & Models (P1) September 5th, 2012

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I. The Graph of an Equation Ex. 1: Sketch the graph of y = (x - 1) 2 - 3.

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You try: Sketch the graph of a. y = (x + 3) 2 + 4 b. y = (x - 2) 2 - 1 c. y = -(x + 1) 2 + 3

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Ex. 2: Use a graphing utility to graph each equation. a. y = x 3 - 3x 2 + 2x + 5 b. y = x 3 - 3x 2 + 2x + 25 c. y = -x 3 - 3x 2 + 20x + 5 d. y = 3x 3 - 40x 2 + 50x - 45 e. y = -(x + 12) 3 f. y = (x - 2)(x - 4)(x - 6)

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II. Intercepts of a Graph Def: The x-intercept of a graph is the point (a, 0) where the graph crosses the x-axis. Find it by plugging y = 0 into the equation and solving for x. The y-intercept is the point (0, b) where the graph crosses the y- axis. Find it by plugging x = 0 into the equation and solving for y.

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Ex. 3: Find the x- and y-intercepts of the graph of y = x 3 - 5x 2 - 6x.

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You try: Find the x- and y-intercepts of the graph of y = 9x 4 - 25x 2.

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Ex. 4: Use a graphing utility to find the x- and y- intercepts of the graph of y = x 4 - 3x 3 + 2x 2 - x - 4.

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III. Symmetry of a Graph Tests for Symmetry: 1. y-axis: Replacing (x, y) with (-x, y) yields an equivalent equation.

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2. x-axis: Replacing (x, y) with (x, -y) yields an equivalent equation.

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3. origin: Replacing (x, y) with (-x, -y) yields an equivalent equation.

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Ex. 5: Test the graph of the equation y = x 3 - 4x for symmetry with respect to each axis and the origin.

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You try: Sketch the graph of the equation 2x + y 2 = 4. Identify any intercepts and test for symmetry.

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IV. Points of Intersection Ex. 6: Find the points of intersection of the graphs of the equations x + 3y = -2 and x 2 - y = 4.

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You try: Find the points of intersection of the graphs of the equations x = y - 2 and x 2 + y 2 = 10.

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