3.5: Graphs of Functions

Determining if a function is even or odd:

In order to determine whether a function is odd even, or neither you must plug in (-x) for the variable x.

Examples:

f(x) = –3x² + 4

f(–x) = –3(–x)² + 4

= –3(x²) + 4

= –3x² + 4

Because the final expression is the same as the original the function is even.

f(x) = 2x³ – 4x

f(–x) = 2(–x)³ – 4(–x)

= 2(–x³) + 4x= –2x³ + 4x

Because the final expression is the opposite of the original the function is odd.

f(x) = 2x³ – 3x² – 4x + 4

f(–x) = 2(–x)³ – 3(–x)² – 4(–x) + 4

= 2(–x³) – 3(x²) + 4x + 4
= –2x³ –3x² + 4x + 4

Because the final expression is neither the same nor opposite the function is neither odd or even.

Types of Function Graphs:

y=f(x)-c - > Vertical Change

y=f(x-c) - > Horizontal Change

y=f(x)*c - > Vertical Stretch (no affects on x-intercepts)

y=f(x*c) - > Horizontal Stretch (no affect on y-intercepts)

This graph illustrates the graph y=|x|

Absolute value function

This graph illustrates the graph y=[x]

Greatest integer function

This graph illustrates a constant function

This graph illustrates y=mx+b

Linear function

This graph illustrates a quadratic function

This graph represents y=x³

Cubic function

This graph represents y= √x

Square root function