3 1 Symmetry and Coordinate Graphs Objectives Use
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 Line Symmetry: Line l P P’ y-axis symmetry 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 = x line and y = -x line. Pg. 129 -130
Symmetric with respect to: x-axis (x, y) → (x, -y) y-axis (also called (x, y) → (-x, y) even) y=x y = -x (x, y) → (y, x) (x, y) → (-y, -x)
Ex. 2: Ex. 3) 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. 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. *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. ) y = x² b. ) c. ) g(x) = -3 x³ + 5 x
Even Functions: Odd Functions: Symmetric with respect to y-axis. f(-x) = f(x) Symmetric with respect to the origin. f(-x) = -f(x)
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