Section 1.7 Symmetry & Transformations Chapter 1 Section 1.7 Symmetry & Transformations

Points and Symmetry

Types of Symmetry Symmetry with respect to the x-axis (x, y) & (x, -y) are reflections across the x-axis y-axis (x, y) & (-x, y) are reflections across the y-axis Origin (x, y) & (-x, -y) are reflections across the origin

Even and Odd Functions Even Function: graph is symmetric to the y-axis Odd Function: graph is symmetric to the origin Note: Except for the function f(x) = 0, a function can not be both even and odd.

Algebraic Tests of Symmetry/Tests for Even & Odd Functions f(x) = - f(x) symmetric to x-axis neither even nor odd (replace y with –y) f(x) = f(-x) symmetric to y-axis even function (replace x with –x) - f(x) = f(-x) symmetric to origin odd function (replace x with –x and y with –y)

Basic Functions

Basic Functions

Basic Functions

Basic Functions

Basic Functions

Basic Functions

Basic Functions

Transformations with the Squaring Function

Transformations with the Absolute Value Function

Transformation Rules Equation How to obtain the graph For (c > 0) y = f(x) + c Shift graph y = f(x) up c units y = f(x) - c Shift graph y = f(x) down c units y = f(x – c) Shift graph y = f(x) right c units y = f(x + c) Shift graph y = f(x) left c units

Multiply y-coordinates of y = f(x) by a Transformation Rules Equation How to obtain the graph y = -f(x) (c > 0) Reflect graph y = f(x) over x-axis y = f(-x) (c > 0) Reflect graph y = f(x) over y-axis y = af(x) (a > 1) Stretch graph y = f(x) vertically by factor of a y = af(x) (0 < a < 1) Shrink graph y = f(x) vertically by Multiply y-coordinates of y = f(x) by a