Increasing and Decreasing Functions Relative Maxima and Relative

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§ § Increasing and Decreasing Functions Relative Maxima and Relative Minima Even and Odd

§ § Increasing and Decreasing Functions Relative Maxima and Relative Minima Even and Odd Functions and Symmetry Functions and Difference Quotients

State the intervals on which the given function is increasing, decreasing, and constant. Constant:

State the intervals on which the given function is increasing, decreasing, and constant. Constant: none Go from left to right like you’re walking a trail. Remember to use x-coordinates.

 Constant: none Go from left to right like you’re walking a trail. Remember

Constant: none Go from left to right like you’re walking a trail. Remember to use x-coordinates.

* (Also known as local maxima and local minima) Keep it simple. These are

* (Also known as local maxima and local minima) Keep it simple. These are the points at which the function changes its increasing or decreasing behavior. They are the turning points.

Where are the relative maxima? Where are the relative minima? Where are the turning

Where are the relative maxima? Where are the relative minima? Where are the turning points?

* The function passes through the origin. The left and right sides of the

* The function passes through the origin. The left and right sides of the graph are reflections of each other. Notice that the y-coordinate stays the same.

 The alternate opposite sides show symmetry to each other. The function passes through

The alternate opposite sides show symmetry to each other. The function passes through the origin.

 It is an even function. It passes through the origin. It is symmetrical

It is an even function. It passes through the origin. It is symmetrical on both sides of the y-axis.

 The function is odd. It passes through the origin. The alternate opposite sides

The function is odd. It passes through the origin. The alternate opposite sides show symmetry to each other.

EVEN FUNCTIONS Algebraically: f(-x) = f(x) for all x in the domain of f.

EVEN FUNCTIONS Algebraically: f(-x) = f(x) for all x in the domain of f. This means you take the function and plug in –x for x. If you end up with the original equation, it is an even function or symmetric with respect to the y-axis.

ODD FUNCTIONS Algebraically: f(-x) = -f(x) for all x in the domain of f.

ODD FUNCTIONS Algebraically: f(-x) = -f(x) for all x in the domain of f. This means you take the function and plug in –x for x. If you end up with the opposite of the original equation – it is an odd function or symmetric with respect to the origin.

Verifying Algebraically….

Verifying Algebraically….

 Odd Even Neither

Odd Even Neither

The Difference Quotient The difference quotient of a function f is an expression of

The Difference Quotient The difference quotient of a function f is an expression of the form where h ≠ 0. Where does it come from? The difference quotient, allows you to find the slope of any curve or line at any single point.

Steps Used in Finding A Difference Quotient

Steps Used in Finding A Difference Quotient