P.3 Graphs of Equations (Part 4)

Symmetry

Symmetry Each of the graphs on the next 3 slides have symmetry with respect to one of the coordinate axes or with respect to the origin

y-axis Symmetry A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph, (-x, y) is also on the graph. y (-x, y) (x, y) x

x-axis Symmetry A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, -y) is also on the graph. y (x, y) x (x, -y)

Origin Symmetry A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph, (-x, -y) is also on the graph. (x, y) (-x, -y)

y-axis Symmetry This graph is symmetric with respect to the y-axis because the point (-x, y) satisfies the equation of the graph. y (-x, y) (x, y) x y = x² - 6

y-axis Symmetry y = x² - 6 y = x² - 6 y = (-x)² - 6 Given equation Substitute (-x, y) for (x, y) Equivalent equation y (-x, y) (x, y) x y = x² - 6

Summary: Tests for Symmetry The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation. The graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation.

Classwork: Testing For Symmetry Worksheet

Homework: Pg 40 Exercises 41-46 (all)

Symmetry Knowing the symmetry of a graph BEFORE attempting to sketch it is helpful. How?? Because then you need only half as many solution points to sketch the graph.

Using Intercepts and Symmetry as Sketching Aids Use intercepts and symmetry to sketch the graph of x - y² = 1. First, let’s see if there are any y-intercepts: Let x = 0. 0 - y² = 1 -y² = 1 or y² = -1 has no real solutions. Therefore, there are no y-intercepts.

Using Intercepts and Symmetry as Sketching Aids Use intercepts and symmetry to sketch the graph of x - y² = 1. Second, let’s see if there are any x-intercepts: Let y = 0. x - 0² = 1 x = 1 Therefore, the x-intercept is (1,0).

Using Intercepts and Symmetry as Sketching Aids Use intercepts and symmetry to sketch the graph of x - y² = 1. Finally, let’s test for symmetry: The only one that satisfies is the test for x-axis symmetry. Therefore, the graph is symmetric with respect to the x-axis.

Using Intercepts and Symmetry as Sketching Aids Use intercepts and symmetry to sketch the graph of x - y² = 1. Now, using symmetry, we can find solution points above the x-axis and then reflect them to get the graph. y 1 2 x = y² + 1 5

Final Graph x - y² = 1 y 1 2 x = y² + 1 5 (5, 2) (2, 1) (1, 0) (2, -1) 1 2 x = y² + 1 5 We know it is symmetric with respect to the x axis, so reflect the points. (5, 2) (2, 1) (1, 0) (2, -1) (5, -2)

Therefore, (0, 1) is the y-intercept. Now try this one: Graph y = |x – 1| First, let’s see if there are any y-intercepts: Let x = 0. y =|0 – 1| = 1 y = 1 Therefore, (0, 1) is the y-intercept.

Therefore, (1, 0) is the x-intercept. Now try this one: Graph y = |x – 1| Second, let’s see if there are any x-intercepts: Let y = 0. 0 =|x – 1| x = 1 Therefore, (1, 0) is the x-intercept.

Now try this one: Graph y = |x – 1| Finally, this equation fails all the tests of symmetry and it is not symmetric to either axis or to the origin. The absolute value sign tells us that y is always positive. x -2 -1 1 2 3 4 y = |x – 1|

Final Graph y = |x – 1| x -2 -1 1 2 3 4 y = |x – 1| (4, 3) (-2, 3) 1 2 3 4 y = |x – 1| (4, 3) (-2, 3) (-1, 2) (3, 2) (0, 1) (2, 1) (1, 0)

Classwork: Symmetry of Graphs Worksheet

Homework: Textbook pg # 40 Exercises: 48, 49, 50, 51, 54, 56, 58, 60, 61, 63, 66, 68 Heads up: Test on Sections P.1 – P.3 on Friday. May use a 1 page cheat sheet.