Example: The graph of x = | y | – 2 shown below, is symmetric to x-axis y x 1 2 –323 A graph is symmetric to x- axis if whenever (x, y) is on graph, so is ( x,- y). The graph intersects the y-axis at (0, 2) and at (0, –2). Symmetry Graphs can have symmetry with respect to the x- axis, the y-axis, or the origin. Symmetry can be confirmed by tables, graphically, or algebraically.

xy(x, y) –23(–2, 3) –12(–1, 2) 01(0, 1) 12(1, 2) 23(2, 3) y x – Example: The graph of y = | x | + 1 shown below, is symmetric to y-axis A graph is symmetric to y-axis if whenever (x, y) is on graph, so is (- x, y). Example: The graph of y = x 3 shown below, is symmetric to origin A graph is symmetric to origin if whenever (x, y) is on graph, so is (- x, -y). y x –22 2 4

Algebraic test for symmetry Symmetric to x-axis –replacing y with –y yields equivalent equation symmetric to y-axis –replacing x with –x yields equivalent equation symmetric to origin –replacing x with –x and y with –y yields an equivalent equation

Example: test 4x + y 2 = 1 for symmetry. x-axis 4x + (-y) 2 = 1 –result 4x + y 2 = 1 y-axis4(-x) + y 2 = 1 –result -4x + y 2 = 1 origin4(-x) + (-y) 2 = 1 –result -4x + y 2 = 1 yes no

Standard equation for circle The equation for a circle with center ( h, k ) and radius r Write the equation of circle with center and radius given (2,3) r = 4 (9,-1) r =7 (-2,-3) r = 5

Given the equation of circle, find the center and radius. (1,-1) r = 5 (-4,0) r =3.1 (-0.2,3) r = 9