3.1 Symmetry and Coordinate Graphs

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3.1 Symmetry & Coordinate Graphs

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Presentation transcript:

3.1 Symmetry and Coordinate Graphs Objectives: Use algebraic tests to determine if the graph of a relation is symmetrical. Classify functions as even or odd.

Group work for lines of symmetry investigation Two distinct points P and P’ are symmetric with respect to a line l iff. l is the bisector of PP’. A point P is symmetric to itself with respect to line l iff. P is on l. *Can be folded in half on line of symmetry and two halves match exactly. -common lines of symmetry are: x-axis, y-axis, y = x line and y = -x line. Pg. 129-130 Line Symmetry: Line l P P’ y-axis symmetry

Symmetric with respect to: x-axis y-axis (also called even) y=x y = -x (x,y) → (x,-y) (x,y) → (-x, y) (x,y) → (y, x) (x,y) → (-y, -x)

Ex. 2: Determine whether the graph of x² + y = 3 is symmetric with respect to the x-axis, the y-axis, the line y = x, y = -x or none of these. Ex. 3) Determine whether the graphs of y = x + 1 is symmetric with respect to the x-axis, the y-axis, both, or neither. Use the information to graph the relation.

Ex. 4)

Symmetric with respect to the origin (odd): A function has a graph that is symmetric with respect to the origin iff. f(-x) = - f(x) for all x in the domain of f. Symmetric with respect to the origin (odd): *Looks the same upside down or right side up. A graph is symmetric with respect to the origin if it is unchanged when reflected across both the x-axis and y-axis. Ex. 1: Determine whether each graph is symmetric with respect to the origin. a.) b.) y = x² c.) g(x) = -3x³ + 5x

Even Functions: Symmetric with respect to y-axis. f(-x) = f(x) Symmetric with respect to the origin. f(-x) = -f(x) Odd Functions: