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.