Graphing Algebraic Functions: Domain and Range, Maxima and Minima
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Graphing Algebraic Functions: Domain and Range, Maxima and Minima


It’s Professor Dave, let’s graph some
functions. We already learned how to construct graphs
when we first introduced some algebraic concepts. Graphing functions will be no different, as
F of X equals two X plus one will be essentially the same as Y equals two X plus one. At its core, graphing is just a visual representation
of a table we could make by plugging in X values, and then recording all of the corresponding
Y values, or values of the function, that result. In this case, when X is one, Y is three. When X is two, Y is five, and so forth. We just graph all of the possible ordered
pairs that comprise the function. To get a little more review on this kind of
stuff, you can go back to some earlier algebra tutorials now, but if you are up to speed,
let’s move on to some other things we can discuss when it comes to graphing functions. The first thing we want to mention is that
as we move through algebra, we won’t always be graphing lines. We will be graphing higher degree functions
as well, meaning there will be X terms that are raised to some exponent. The simplest of these is the parabola. Let’s look at F of X equals X squared. Plugging in zero gives us zero, so the origin
is part of this function. Then as we move to the right, the slope continues
to increase. When X is one, we get one. When X is two, we get four. We get the same thing in the negative direction. That means that in this particular case, the
function goes to positive infinity as X approaches negative or positive infinity. We can look at what even higher degree functions
do as well. Here is X cubed. Now the function grows even faster, but we
must note that when we cube negative numbers, we get negative numbers, so as X goes to negative
infinity, the function goes to negative infinity, rather than positive infinity. If we look at X to the fourth, it goes back
to looking like X squared, just steeper. In this way, we can make the generalization
that for a polynomial with one term, if that term has an odd exponent, it is an odd function,
and it will somewhat resemble the X cubed function. If the exponent is even, it is an even function,
and it will somewhat resemble the X squared function. Another thing we must frequently describe
is the domain of a function. As we mentioned before, this is the set of
all allowed X values. Very frequently, the domain will be all real
numbers. This is because for most lines and curves,
any X value can be plugged in, and we can see this represented visually with a graph
that extends infinitely in both horizontal directions. Exceptions would be things like a vertical
line. Here, only one X value is in the domain. There are also instances where a sum or difference
in the denominator of a function rules out a specific X value so as to avoid dividing
by zero, as this would produce undefined values. This is where we get asymptotes, whereby a
value is missing from the domain, and the function will typically go towards positive
or negative infinity as it approaches this missing value. More on those later. By contrast, the range of a function, or the
set of allowed values for the function, can vary dramatically. Take something like X squared. While you can plug in any X value, the function
only has values that are greater than or equal to zero. Something like a horizontal line would have
only one value for its range. But many functions will have a range of all
real numbers, if they extend from negative infinity to positive infinity, the way X cubed
does. Beyond these parameters, we can identify relative
maxima and minima of a function, which are the hills and valleys. A relative maximum is a point where the function
changes from increasing to decreasing, and a relative minimum is a point where the function
changes from decreasing to increasing. Many of these concepts will come back later,
but for now, let’s check comprehension.

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