0-2: Smart Graphing Objectives: Identify symmetrical graphs Identify odd/even functions Sketch the graphs of functions using translations, reflections & dilations © 2002 Roy L. Gover (roygover@att.net) Modified by Mike Efram 2004

Definition Point Symmetry: Two points, P & P’ are symmetric with respect to a point M if M is the midpoint of M P P’

...For a graph to have point symmetry with respect to a point M, M must be the midpoint of every set of points P & P’ on the graph. Examples...

Example Point Symmetry Consider:

Example Point Symmetry: M M

Definition A graph that is symmetrical with the point (0,0) is symmetric with respect to the origin.

Definition A function f(x) is symmetric with respect to the origin if and only if f(-x)=-f(x)

Example f(x)=x3 is symmetric with the origin because f(-x)=-f(x). ie f(-2)=-8 & f(2)=8,therefore f(-2)=-f(2)

Is f(x)=x2 symmetric with respect to the origin? Try This Is f(x)=x2 symmetric with respect to the origin? No

Important Idea Graphs that have line symmetry can be folded along the line of symmetry so that the two halves match exactly.

Examples of Line Symmetry

Symmetry with respect to x=0 ( y-axis ) exists if and only if: Definition Symmetry with respect to x=0 ( y-axis ) exists if and only if: f(x)=f(-x) Example: f(x)=x2-3

Important Idea Symmetry is useful in graphing functions. If you graph part of the function and understand the symmetry, the rest of the graph can be sketched.

Definition Even Functions are functions symmetric with the y axis. They have exponents that are all even.

Definition Odd functions are functions symmetric with the origin. They have exponents that are all odd.

Try This Are the following functions even, odd or neither: Even Odd

Summary Odd functions: f(-x) = -f(x) Symmetry with origin (0, 0)

Summary Even functions: f(x) = f(-x) Symmetry with y-axis

Definition Reflections: the mirror image of a graph. Example f(x)=x2

Try This Without using a graphing calculator, graph f(x)=-x3 using its parent graph as a starting point.

Solution

Definition Translation: the sliding of a graph vertically or horizontally without changing its size or shape.

Examples Vertical Translations Horizontal Translations f(x)=x2+3

Try This Write the equation of this graph based on its parent graph. Hint: a vertical & horizontal translation is required.

Try This Write the equation of this graph based on its parent graph. Hint: a reflection & horizontal translation is required.

Try This Without using your calculator, sketch the graph of:

Definition Dilation: changing a graph’s size. Making it either smaller or larger. Examples:

Example The graph of f(x) is pictured at the right. Sketch a graph of: b) f(x+3)-2 c) -f(x-3)-2 d) 2f(x+2)+3 a) f(x+3)