Graphs & Models (P1)

September 5th, 2012

I. The Graph of an Equation

•Ex. 1: Sketch the graph of y = (x - 1)2 - 3.

You try:

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

Ex. 2: Use a graphing utility to graph each equation.

a. y = x3 - 3x2 + 2x + 5b. y = x3 - 3x2 + 2x + 25c. y = -x3 - 3x2 + 20x + 5d. y = 3x3 - 40x2 + 50x - 45e. y = -(x + 12)3

f. y = (x - 2)(x - 4)(x - 6)

II. Intercepts of a GraphDef: 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.

Ex. 3: Find the x- and y-intercepts of the graph of y = x3 - 5x2 - 6x.

You try:

Find the x- and y-intercepts of the graph of y = 9x4 - 25x2.

Ex. 4: Use a graphing utility to find the x- and y-intercepts of the graph of y = x4 - 3x3 + 2x2 - x -

4.

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

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

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

Ex. 5: Test the graph of the equation y = x3 - 4x for symmetry with respect to each axis and the origin.

You try: Sketch the graph of the equation 2x + y2 = 4. Identify any intercepts and test for symmetry.

IV. Points of Intersection

Ex. 6: Find the points of intersection of the graphs of the equations x + 3y = -2 and x2 - y = 4.

You try: Find the points of intersection of the graphs of the equations x = y - 2 and x2 + y2 = 10.