4.3 Symmetry Objective To reflect graphs and use symmetry to sketch graphs. Be able to test equations for symmetry. Use equations to describe reflections and translations of graphs.

Symmetry A line l is called an of a graph if it is possible to pair the points of the graph in such a way that l is the perpendicular bisector of the segment joining each pair. axis of symmetry l is the axis of symmetry A point O is called of a graph if it is possible to pair the points of the graph in such a way that O is the midpoint of the segment joining each pair point of symmetry O is the point of symmetry

Symmetry A line l is called an of a graph if it is possible to pair the points of the graph in such a way that l is the perpendicular bisector of the segment joining each pair. axis of symmetry l is the axis of symmetry A point O is called of a graph if it is possible to pair the points of the graph in such a way that O is the midpoint of the segment joining each pair point of symmetry O is the point of symmetry

Special Tests for the Symmetry of a Graph Type of SymmetryTesting an equation Symmetric with respect to the x-axis Symmetric with respect to the y-axis (x, y) Symmetric with respect to the line y = x (x, y) Symmetry in the origin (x, y) Substitute (x, y) with (x, -y) & (-x, y) are on the graph & (-x, -y) are on the graph & (y, x) are on the graph (x, y) & (x, -y) are on the graph Substitute (x, y) with (-x, y) Substitute (x, y) with (y, x) Substitute (x, y) with (-x, -y)

Symmetric with respect to?? ORIGIN

Symmetric with respect to?? X-AXIS

Symmetric with respect to?? Y-AXIS

Use the graph to tell the new location of the given point after it is reflected over the specified line or origin. A B 1.A; x-axis 2.B; y-axis 3.A; y = x 4.B; origin (2, -3) (3, 1) (3, 2) (3, -1)

Even and Odd Functions A function f is an even function if and only if 1. The domain of f is symmetric about the origin; and 2. f(– x) = f(x) for all x in the domain. **The graph of an even function has symmetry in the y-axis. A function f is an odd function if and only if 1.The domain of f is symmetric about the origin; and 2. f(– x) = –f(x) for all x in the domain. **The graph of an odd function has symmetry in the origin.

More on Graphs of Even and Odd Functions Symmetry in the y axis Even Function Symmetry in the origin Odd Function

What type of symmetry does each function have? Then state if the graph is even, odd, or neither xy 2 = 1 X-Axis:y = x Y-axis:Origin:

y = x 2 X-Axis:y = x Y-axis:Origin: What type of symmetry does each function have? Then state if the graph is even, odd, or neither

x 3 + y 3 = 1 X-Axis:y = x Y-axis:Origin: What type of symmetry does each function have? Then state if the graph is even, odd, or neither

y = x 3 X-Axis:y = x Y-axis:Origin: What type of symmetry does each function have? Then state if the graph is even, odd, or neither

x 4 + y 4 = 1 X-Axis:y = x Y-axis:Origin: What type of symmetry does each function have? Then state if the graph is even, odd, or neither

Homework: Textbook p. 136 #15, 16, 31