# Describing Graphs of Polynomial Functions

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*turning points,*it's possible to give a much more detailed description of polynomial graphs.

## Increasing and Decreasing Intervals

A function is said to be increasing when, as the $x$-values increase (from left to right), the values of $f(x)$ **increase**. Conversely, a function is said to be decreasing when, as $x$ increases, $f(x)$ **decreases**. The graph below shows increasing intervals with green arrows and decreasing intervals with red arrows.

The function above contains two increasing intervals and one decreasing interval. To describe each, use the $x$-values. Commonly a point where a function has a relative maximum or a relative minimum is neither included in an increasing nor a decreasing interval.

$\begin{aligned} \textbf{Increasing interval:} & \ \text{-} \infty < x < \text{-} 2 \ \text{and} \ 0 < x < + \infty \\[0.8em] \textbf{Decreasing interval:} & \qquad \qquad \ \text{-} 2 < x < 0 \\ \end{aligned}$## Relative Minimum and Maximum

For some functions, their graphs extend infinitely in the vertical direction. For these graphs, there are no highest or lowest points. However, it's possible for these functions to have relative minimum or relative maximum values. A relative minimum is the lowest point for a region of the graph. Similarly, a relative maximum is the highest point for a region of the graph.

## Turning Point

Relative minimums and maximums of polynomial functions are also called turning points. This is because a function's graph turns from increasing to decreasing, or vice versa, at these points.

The graph of a polynomial function of degree $n$ can have **at most** $n - 1$ turning points. The function shown is a $3$rd degree polynomial and has $2$ turning points.

**at least**$n - 1$ turning points.

## Even and Odd Symmetry

If a function has a symmetry, it is either *even* or *odd*. The symmetry is even when the graph is symmetric with respect to the $y$-axis, and odd when it's symmetric about the origin.

### Even Symmetry

If a function has even symmetry, the following rule applies: $f(\text{-} x)=f(x).$ The rule comes from the fact that even symmetry is a reflection across the $y$-axis. Therefore, changing the sign of the $x$-value does not affect the function value.

The concept applies both ways. Hence, if the rule is true for the entire domain, the function has even symmetry.

### Odd Symmetry

Instead, if a function has odd symmetry, the rule it must follow is $f(\text{-} x)=\text{-} f(x).$ An odd symmetry means graphically that the graph is rotated $180^\circ$ about the origin. Therefore, changing the sign of the $x$-value also changes the sign of the function value.

If this rule is satisfied on the entire domain, the function has odd symmetry.## Exercises

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