Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes. In 1.1, we discussed using t-charts to help graph functions. e.g.

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1 Accelerated Precalculus 1.2 (Intercepts and Symmetry) Day 1 Notes In 1.1, we discussed using t-charts to help graph functions. e.g., Graph: y = x 3 What are some other strategies that can make graphing simpler? Let s take a look at one of the graphs from your homework problems on 1.1: Name the x-intercepts. Name the y-intercepts. What do you notice about the coordinates of the intercepts?

2 Therefore, the procedure for finding intercepts algebraically from an equation: To find x-intercept(s), if any, let y = 0 in the equations and solve for x, where x R. To find y-intercept(s), if any, let x = 0 in the equations and solve for y, where y R. Example 1 Find the x-intercept(s) and the y-intercept(s) of the graph of y = x 2 4. Then, graph the function by plotting points. Another helpful tool for graphing equations by hand involves symmetry. Symmetry often occurs in nature. Consider the starfish below. Do you see the symmetry?

3 What do you notice about the symmetry in each graph below?

4 Let s generalize A graph is said to be symmetric with respect to the x-axis if, for every point (x, y) on the graph, the point (x, y) is also on the graph. A graph that is symmetric about the x-axis is not the graph of a function (except for the graph of y = 0). A graph is said to be symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point ( x, y) is also on the graph. A graph that is symmetric about the y-axis is known as an even function. A graph is said to be symmetric with respect to the origin if, for every point (x, y) on the graph, the point ( x, y) is also on the graph. A graph that is symmetric about the origin is known as an odd function. Symmetry with respect to the origin may be viewed in three ways: As a reflection about the y-axis, followed by a reflection about the x-axis As a reflection about the x-axis, followed by a reflection about the y-axis As a projection along a line through the origin so that the distances from the origin are equal

5 Example 2 Fill in the blanks: (a) If a graph is symmetric with respect to the x-axis and the point (4, 2) is on the graph, then the point is also on the graph. (b) If a graph is symmetric with respect to the y-axis and the point (4, 2) is on the graph, then the point is also on the graph. (c) If a graph is symmetric with respect to the origin and the point (4, 2) is on the graph, then the point is also on the graph. How can we test for symmetry? Tests for Symmetry To test the graph of an equation for symmetry with respect to the x-axis Replace y by y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the x-axis. y-axis Replace x by x in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the y-axis. Origin Replace x by x and y by y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.

6 Example 3 For the equation y = x2 4 x 2 +1, (a) Find the intercepts. (b) Test for symmetry.

7 When the graph of an equation is symmetric with respect to the x-axis, the y-axis, or the origin, the number of points that you need to plot in order to see the pattern is reduced. For example, if the graph of an equation is symmetric with respect to the y-axis, then once points to the right of the y-axis are plotted, an equal number of points on the graph can be obtained by reflecting them about the y-axis. Because of this, before we graph an equation, we first want to determine whether it has any symmetry. Homework: 1.2 (p ) #3-10 (all), 14, 18, 22, 24, 27, 28, 30, (odds), (all), and (odds)

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9 1.2 (p ) #3-10 (all), 14, 18, 22, 24, 27, 28, 30, (odds), (all), (odds) Answers 3. intercepts 4. y-axis ( 3, 4) 7. true 8. False 9. a 10. c (a) ( 1, 0), (1, 0) (b) sym. w.r.t. x-axis, y-axis, and origin 35. (a) ( π 2, 0), (0, 1), (π 2, 0) (b) sym. w.r.t. the y-axis 37. (a) (0, 0) (b) sym. w.r.t. the x-axis 39. (a) ( 2, 0), (0, 0), (2, 0) (b) sym. w.r.t. the origin 41. (a) (x, 0), 2 x (a) no intercepts (b) no symmetry (b) sym. w.r.t. the origin

10 ( 4, 0), (0, 2), (0, 2); symmetric w.r.t. the x-axis 51. (0, 0); symmetric w.r.t. the origin 53. (0, 9), (3, 0), ( 3, 0); symmetric w.r.t. the y-axis 55. ( 2, 0), (2, 0), (0, 3), (0, 3); symmetric w.r.t. the x-axis, y-axis, and origin 57. (0, 27), (3, 0); no symmetry 59. (0, 4), (4, 0), ( 1, 0); no symmetry 61. (0, 0); symmetric w.r.t. the origin 63. (0, 0); symmetric w.r.t. the origin