Graphing Theory

Introduction §

I noticed that I’ve had some trouble with graphs recently, which really shouldn’t be happening considering I am a Big Grown Up Math Guy. So, here is some review and me messing around with graphing trying to figure out a way to fully understand graphing any type of manipulation and up to any power.

This key skill is something we lose as we go progress into primarily working with equations as opposed to primarily working with graphs. I myself have noticed that it is much harder than it used to be for me to graph without heavily reliance on technology and calculators. This is not in the curriculum so if you somehow come across this it was 100% for fun lol.

Graphing §

Flatness §

Consider the graphing below of $x^4$ and $x^2$. $x^4$ is in blue, $x^2$ is in red. Both of these functions do have some similarities:

• Neither function crosses the $x$-axis
• Both functions are symmetrical across the $y$-axis
• ${\lim }_{x\to \pm \infty }=\infty$ for both

However, the most notable difference between the graph of $x^4$ and the graph of $x^2$ is that $x^4$ grows at a far faster rate than $x^2$ does. $x^4$ is steeper on average than $x^2$, so less of it is ‘on-screen’ because its $y$-values (output) outpace the growth of its $x$-values (input) so much and so quickly that it outgrows the constrains of our medium before $x^2$ does.

This is pretty obvious, considering that $x^4$ is just $(x^2{)}^2$— every output from $x^2$ could be mapped to the output with the same input from $x^4$ by just squaring it. Thus, $x^4$ is a transformation of $x^2$. Let’s look at the growth of $x^4$ around $(0,0)$, on both sides. Before $x^4$ starts shooting up, it appears to be almost flat, or at least more flat for more time than $x^2$ in the same region, though $x^2$ is also growing slower around there. Though you’ve almost certainly caught onto this by now, it is because the squaring (or cubing, or raising to any power really…) of a decimal produces a smaller number.

Sometimes people have trouble with this. It’s easier to thing of it in terms of percents when you just can’t draw that line conceptually. 40% of 40% is smaller than 40%. 40% as a decimal is $0.4$; $(0.4\cdot 0.4)<0.4$; $(0.4{)}^2<0.4$. So $0<x<1$, $x^4$ will give us an output that is lower in value than that of $x^2$, unlike the behavior it exhibits everywhere else. That also means that as we keep raising and raising, it’s just going to get flatter and flatter.

Addition and Subtraction §

We know from earlier experience with graph that subtracting a constant from a function will yield a transformation to the right or to the left. The blue line represents $y=x$; the red one is $y=x-1$. But when we want to start subtracting functions from functions or whatever you want to call it, we have to consider that when we’re raising to a power more than one things happen exponentially, not linearly. So, we can set up something with the equations we have above, and analyze the differences.

$$y=x^4-x^2$$

Let’s say we’re trying to find the output of $y$ in at $x=5$. Let’s demonstrate the concept by plugging in 5 for some other $x^4-$ x-raised-to-some-power equations, so that we can see it in action.

$$\begin{array}{rl}x& =5\\ x^4-x=(5{)}^4-5& =620\\ x^4-x^2=(5{)}^4-(5{)}^2& =600\\ x^4-x^3=(5{)}^4-(5{)}^3& =500\end{array}$$

Bla bla bla obviously the value of the second term decreases exponentially, we see it go down.

$$\begin{array}{rl}x& =.5\\ x^4-x=(0.5{)}^4-0.5& =-0.4375\\ x^4-x^2=(0.5{)}^4-(0.5{)}^2& =-0.1875\\ x^4-x^3=(0.5{)}^4-(0.5{)}^3& =-0.0625\end{array}$$

We can see that despite the fact that $x^4$ and $x^2$ do not cross the $x$-axis, $x^4-x^2$ does because $x^2$ ends up bigger than $x^4$ in our special $0<x<1$ zone. This means that we get two parts of the graph, one on either side of the $y$-axis, where we briefly dip under the $x$-axis and into the negative zone. While the magnitude of the numbers in this area are going to be small, we can still see the ‘flatness!” We also observe an interesting effect when we look at the graphs of $x^4$ (red) and $x^2$ (green) compared to $x^4-x^2$ (blue). You can actually visibly observe that in our Magic Range (on both sides, but since this one is symmetrical it is not super important) $x^2$ outpaces $x^4$ until we reach the point $(1,1)$, where $x^4=x^2$. After this point they switch places.

In the areas where $x^4<x^2$, we dip under the $x$-axis; otherwise, we find a graph that more closely follows the original $x^4$ because it is exponentially bigger than $x^2$ from this point on. if you put a kid who weight 9 $(3^2)$ kg on one side of a seesaw and a bigger guy who weighs 81 $(3^4)$ kg on the other side, sure, you’ll get some influence on the balance of the seesaw from the kid, but it’ll mostly be from the other guy.

As we get to working with bigger and bigger values, this difference becomes more pronounced. How about a 16 $(4^2)$ kg kid with a 256 $(4^4)$ kg obese dude on the other side? That difference is bigger than the difference between 81 and 9. $256-16=240$ is closer to 256 than $81-9=72$ is to 81 proportionally— $\frac{240}{256}\approx 0.94$ while $\frac{72}{81}\approx 0.89$. So it stands to reason that the blue line will stick to the red line even more closely as we zoom out.

Just as we shifted graphs in our formative algebra lessons, we’ve actually just shifted the graph of $x^4$ down by $x^2$— it’s just that $x^2$ is different at every point, so we need to thing a little more about its behavior. It is easy to be fooled and think that the red graph of $x^4$ is “smaller” than the blue graph of $x^4-x^2$, but in reality it is growing quicker— that’s why it is on the inside, it doesn’t take as long for it to get to business.

It is also important to note that because the graph ends up adhering so closely to the graph of $x^4$ as ${\lim }_{x\to \pm \infty }{x}^4-x^2$, the difference in their slopes approaches zero. But even if the slopes are practically the same, $x^4-x^2$ is still shifted down by $x^2$… so it’ll never catch up. Addition is usually easier than subtraction, so we actually have the hard part out of the way right now. It’s the same concept, really, except this time instead of considering the behavior of the graphs when subtracting we’ll consider then when we’re adding.

Let’s review what we originally said was in common with the graphs of $x^4$ and $x^2$ again. We’ll pull up the same image as before, of Graph A: Okay, so what do we know from the similarities we outlined before? How can we apply this in the context of $y=x^4+x^2$ this time?

• Neither function crosses the $x$-axis This means that neither of these functions ever become negative. A positive real number plus a positive real number can never equal a negative, so the graph of $y$ will also not cross the $x$-axis and it will remain in the top two quadrants.
• Both functions are symmetrical across the $y$-axis If every output is the same ever for the absolute value of every input, then we will get the same $y$-values for $x$-values that are of equivalent distance from the origin. Aka, it’s still symmetrical— nothing interrupts this symmetry.
• ${\lim }_{x\to \pm \infty }=\infty$ for both Again, it’s the same. While some infinities are bigger than others, that is a whole other tangent. Infinity plus infinity is infinity. The new graph will have the same trait.

We can also consider the traditional way we briefly touched on above. We’re shifting the graph of $x^4$ up by a factor of $x^2$. Though $x^2$ differs each time, every increment makes the graph of $y$ closer to the graph of $x^4$, though this time $y$ is actually growing faster than $x^2$. Though it still becomes negligible as we approach infinity, $x^4$ will never actually be equal to $y$ because $y$ gets that cheat of being shifted up $x^2$ to start with.

Okay, so let’s put it all together— what is the graph going to look like?

1. It will not cross the $x$-axis
2. It will be symmetrical.
3. ${\lim }_{x\to \pm \infty }y=\infty$
4. It will be greater than $x^4$ and $x^2$, but as it grows it will adhere closer to the graph of $x^4$ than $x^2$
5. It will be less flat than both $x^4$ and $x^2$— i.e., the corresponding $y$-values of $x$-values with magnitudes less than one will be bigger, but it will be closer to the graph of $x^2$ in this area because it is more ‘heavy’ in this area.
6. It will intersect the origin at $(0,0)$ and be centered around this point.
7. It will resemble a parabola. Hey, that’s a job well done. Let’s quickly inspect the area around the origin, too, since we’re already here… Once again we are correct here— if you add two positive values together, you can only get a bigger value out of it.

Multiplication §

What if, instead of this adding and subtracting nonsense, we wanted to find the graph of a function like $y=x^3$? If you already know what $x^3$ looks like, that doesn’t really matter— we’re going to figure this one out, then apply this method further to get some dumb looking graphs.

Transformation Review §

Before you even think about graphing this, remember what happens when we multiply a parabola by a constant. We can use $f(x)=5x^2$ as an example. Though what we’re multiplying by stays the same here, the distance between each value as we go on just keeps getting larger.

x	f(x)	x^2
1	5	1
2	20	4
3	45	9
4	80	16
5	125	25
6	180	36

The distance— and therefore the slope— continues to increase, and it is increasing faster for f(x) than it is for $x^2$ because of that pesky little constant. I.e., f(x) is steeper than $x^2$. BTW — $f^{\prime }(x)=10x$ models the slope for that whole parabola. Derivatives are everywhere, baby… everywhere.

Graphing $x^3$ §

While there are a lot of ways to graph multiplication between terms, things like this are often overlooked. $y=x^3$ is actually just $y=x(x^2)$. Yes, we’re allowed to know what those look like. I don’t make the rules, I just think them up, write them down and enforce them. Anyway, you can see these notorious bad boys below. Behavior of $x^2$:

• Does not cross $x$-axis
• Output is always positive
• Symmetrical across $y$-axis — $y$ in $(-x,y)$ equal to $y$ in $(+x,y)$
• Hits the point $(0,0)$
• Hits the point $(1,1)$
• ${\lim }_{x\to \pm \infty }=\infty$
• Slope increases as magnitude of $x$ in increases
• Slightly flat around origin

Behavior of $x$:

• Crosses $x$-axis
• Output can be positive or negative
• Not symmetrical across $y$-axis
• Hits the point $(0,0)$
• Hits the point $(1,1)$
• ${\lim }_{x\to \infty }=\infty ,{\lim }_{x\to -\infty }=-\infty$
• Slope is constant
• Not flat around origin

Now, imagine the behavior of $x^3$ based off of the behaviors above, since we are just multiplying the outputs of these equations. Let’s go through each ‘step’ and identify the similarities and differences, and what this means for our final graph.

$x^2$ does not cross the $x$-axis, while $x$ does. When the graph of a function crosses the $x$-axis, this just means that the $y$-value or output at this point is negative. Because $x^2$ does not cross the $x$-axis, none of its outputs are negative. $x$, on the other hand, is negative when its input is negative— i.e. when $x<0$.

Negative times positive equals negative, so we know that the output of $y=x^3$ will be negative when the input is negative. The opposite is how we know that it will be positive when the input is positive. The basics of raising a function to the third power is some real rocket science, I know.

This also makes it so that the function is not symmetrical across the $y$-axis, but the pattern does repeat— it just repeats in the opposite direction and in the diagonally across quadrant, kind of like the line of $x$. Since $0\times 0=0$ and $1\times 1=1$, we know that $x^3$ will still pass through $(0,0)$ and $(1,1)$.

Building on the previously mentioned sign conservation concept thingamabob, we know that ${\lim }_{x\to -\infty }{x}^3=-\infty$ and ${\lim }_{x\to \infty }{x}^3=\infty$.

The slope will always be increasing. We can pretty quickly find that $\frac{d}{dx}\left[x^3\right]=3x^2$ using basic derivative rules, but if for some reason we couldn’t or it got more annoying to do it that way, we can find it using the slopes of the component functions.

$$\begin{array}{rl}\frac{d}{dx}\left[x^2\right]& =2x\\ \frac{d}{dx}\left[x\right]& =1\\ \frac{d}{dx}[x^3]& =\left(\frac{d}{dx}\left[x\right]\cdot x^2\right)+\left(\frac{d}{dx}\left[x^2\right]\cdot x\right)\\ & =x^2+2x^2\\ & =3x^2\end{array}$$

We know that not only is the slope not constant but that it increases exponentially, making $x^3$ steeper in general than $x^2$ and $x$. This means that the position of $x^3$ relative to that of the other two functions is ‘outermost’ when $|x|<1$ and ‘innermost’ when $|x|>1$, and that it crosses over from outermost to innermost at $x=1$. This also means, based off of what we know about flatness around the origin, that $x^3$ will be flatter around the origin than $x^2$ and $x$ because there are more repetitions.

Here is all of that information in a slightly abridged version of our mini-template from above:

• Crosses $x$-axis
• Sign of output is equal to sign of input
• Not symmetrical across $y$-axis
• Hits the point $(0,0)$
• Hits the point $(1,1)$
• ${\lim }_{x\to \infty }=\infty ,{\lim }_{x\to -\infty }=-\infty$
• Slope is not constant— it is equal to $3x^2$, which is larger than the slope of $x^2$
• Flatter than the functions from before about the origin
• Crosses from being “outside” of $x^2$ and $x$ to being “inside” of them at $|x|=1$

So… let’s graph!