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==Introduction== Graphing functions is not that different from graphing equations. The biggest difference is that for each value of x in t..."

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==Introduction==
Graphing functions is not that different from graphing equations. The biggest difference is that for each value of x in the domain we have only one y-value. In addition to graphing functions,
we will discuss being able to identify whether or not a graph is the graph of a function. We will also discuss symmetries in the graph of a function.

==Vertical Line Test==
Given a set of points in the xy-plane, the points form the graph of a function if and only if the set of points intersects any vertical line in at most one point. This is called the vertical line test.

Non-example: A circle fails to be the graph of a function since the vertical diameter intersects the graph of the circle in two points.

==Symmetry==
Just as we have discussed the symmetry in the graph of an equation, we can discuss the symmetry in the graph of a function.

We call a function even if its graph is symmetric across the y-axis. In terms of the function definition, this means f(-x) = f(x).

We call a function odd if its graph is symmetric with respect to the origin. Another way to express this to to say that f(-x) = -f(x).

Some basic examples of even functions are <math> x^2, \, x^4</math>, while some examples of odd functions are <math> x, x^3</math>

==Increasing, Decreasing or Constant==
We say a function is increasing(decreasing) if the value of the function is increasing(decreasing) as the x-value increases.
A function is constant if it is neither increasing nor decreasing.

Using these definitions we can talk about the intervals over which a funciton is increasing, decreasing, or constant.

A more formal way to express these three concepts is as follows:

1) A function is increasing on an open interval I if, <math>f(x_1) > f(x_2)</math> for any <math> x_1 > x_2,\text{ and } x_1, x_2 \text{ in I.}</math>

2) A function is decrasing on an open interval I if, <math>f(x_1) < f(x_2)</math> for any <math> x_1 < x_2,\text{ and } x_1, x_2 \text{ in I.}</math>

3) A function is constant on an open interval I if, <math>f(x_1) = f(x_2)</math> for any <math> x_1 \neq x_2,\text{ and } x_1, x_2 \text{ in I.}</math>

==Maxima and Minima==

Now that we can identify intervals where the function is increasing and decreasing, we can discuss local maxima/minima, which are points where the function switches from increasing to decreasing, or vice versa.
For an example you can think of the apex (highest point) a tossed ball reaches. Once the ball reaches its apex it stops going higher and starts falling.

We now follow with a more formal definition:

A function f has a local maxima(minima) at a point c if there is an open interval containing c such that <math> f(x) \le f(c), f(x) \ge f(c)\text{ respectively.}</math> In that case we call f(c)
a local maxima(minima) value of f.

If a local maxima(minima) is larger(smaller) that the function value at other point, then we call the value the absolute maximum(minimum).

This leads us to the Extreme Value Theorem:

If a f is a continuous function on a closed interval [a, b], then f has an absolute maximum and an absolute minimum on [a, b]

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MathAdmin: Created page with "__TOC__ ==Introduction== Graphing functions is not that different from graphing equations. The biggest difference is that for each value of x in t..."

MathAdmin: Created page with "
TOC
==Introduction== Graphing functions is not that different from graphing equations. The biggest difference is that for each value of x in t..."