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Some functions are less expressive than others. We're not trying to be judgey, here: it's true. For example, take the graph of a linear function. (We like x = 4; there's a lot of history there.) Draw your line on a coordinate plane, step back, and squint. It's so linear. Simple. Predictable.
Look, we're not saying it's boring, but...yeah. It kind of is.
In this section, we'll be taking a look at quadratic and exponential functions before banking a sharp left and exploring the wonderful world of inequalities. Trust us, things are about to get a lot more exciting around here.
At least in algebra terms, anyway.
We know what quadratic equations are. Now we're going to graph them. That's just how we roll. Whenever we find out what the Mega Millions lottery numbers are, we graph those, too. Same goes for basketball scores, election results, and opening weekend movie grosses. We might have a problem.
In order to get functions, we'll graph quadratic equations of the form
y = ax2 + bx + c.
This will ensure that we only have one value of y for each value of x.
Graph the equation y = x2.
This is the simplest quadratic equation there is. It's also a relation, so we already know how to graph it. We may have forgotten it, but the knowledge is in there somewhere.
First we make a table of values so we can graph some points, and then we see the shape we're getting and connect the dots. Here's a sampling of points:
Now let's graph 'em:
This is forming a sort of "U'' shape, so that's how we connect the dots:
The only root of the polynomial x2 is x = 0. Therefore, 0 is the only value of x we can plug into the equation y = x2 if we want to get 0 for y. Un-coincidentally, the only point on the graph of y = x2 with a y-coordinate of 0 is the point (0, 0).
Once again, an appearance by the ever-popular 0. That guy is everywhere lately. He must be promoting a new film.
Let's do some slightly more interesting, though more complicated, examples. Of course, we could just graph these on the calculator, but we'd like to understand why we're getting the pictures we're getting. So unless you have one of those calculators that blabs on and on about how it "got there, " let's go ahead and do the work ourselves.
Sometimes we'll be asked to sketch a graph. Don't stress; no points will be taken off for less-than-perfectly-straight lines or a complete lack of artistic ability. The idea is to draw the rough shape of the graph and label a couple of easy values, but not to worry about pinpoint accuracy. Save the pinpoint accuracy for archery practice.
Without using a calculator, sketch a graph of the function y = x2 + 1.
Honestly, when we graph y = x2 + 1, we get the graph of y = x2 moved up by 1 unit. The only value we need to label on the sketch is y = 1.
The examples we've done so far have been pretty straightforward, but now we'll get into some examples that are straightbackward. The graphs in our prior examples looked like the graph of x2 moved up or down a bit. To graph general quadratic equations, however, we need to do things differently.
We need a few new definitions first, so spread open your mind and clear some room...
The graph of any quadratic equation is a parabola. A parabola will look like either a right-side-up "U'':
...or an upside-down "U'':
The places the parabola crosses the x-axis are called the x-intercepts, just like with linear equations. The place the parabola crosses the y-axis is the y-intercept. The lowest or highest point of the parabola, depending on which way it opens, is called the vertex of the parabola. Not a vortex, so no need to worry about being swallowed up into a whirling, tornado-like spiral...unless you live in Kansas.
In order to sketch the graph of a general quadratic equation, we need to know three things.
Let's walk through an example to see how to find all these things, and how to put them together into a graph. This will be a nice, relaxing walk. You can even do it sitting down; that's how relaxing it is.
Sketch a graph of y = x2 + 3x + 2.
We need to find the intercepts, the vertex, and whether the parabola opens upwards or downwards. We should also find out how late it's going to be open, in case we need to make a late night run.
1. Where does the parabola cross the axes?
First, where does the parabola cross the x-axis? An x-intercept is a point of the form (something, 0).
Therefore, the x-intercepts of our graph will occur at whatever values of x make y zero; in other words, at the roots of the polynomial x2 + 3x + 2. To find the roots, we factor the polynomial, set it equal to zero, and solve for x.
x2 + 3x + 2 = 0
(x + 2)(x + 1) = 0
x = -2, x = -1
These are the roots of the polynomial, and as a bonus, they're also the x-intercepts of the parabola. It's not a problem that they're both negative. This is an equal opportunity graph.
We now know the points (-2, 0) and (-1, 0) are on the graph:
Where does the parabola cross the y-axis? To find the y-intercept, we plug in 0 for x and see what we get. In this case, we find:
y = (0)2 + 3(0) + 2 = 2
So the y-intercept is 2. We also have the point (0, 2) on the graph:
We now have two points. If this were the NFL, that'd be a safety.
2. What is the vertex?
The vertex of a parabola occurs halfway between the roots, at least when the roots exist. We'll worry in a moment about what happens when they don't. Well, you can worry about it now, but please keep it to yourself until we get to it. No good can come of spoiling it for everyone else in the meantime.
For this parabola, then, the vertex occurs when x is halfway between -2 and -1, or at . We find the y-value by plugging into the quadratic equation:
Nice. The point is also on our parabola.
3. Does the graph open up or down?
Ever tried to push a "pull" door? Because we, um, haven't. It's important to know which way things open. A parabola is no different.
When x is outside the x-intercepts, the further away x gets from zero, the larger y gets, and it's not even taking growth hormones. When x = 5, y = 42; when x = 100, y = 10,302. We can imagine that if we graphed more points, we'd see the graph opening upwards.
Putting all the pieces together, we connect our dots in a "U'' shape, like this:
Now that we've gone through a sample problem and you enjoyed it so much that your mouth is in the shape of an upward-opening parabola, let's talk a little more about the sub-problems involved in graphing a quadratic equation of the form y = ax2 + bx + c.
1. Finding the intercepts.
There are two steps here: finding the x-intercepts and finding the y-intercept. It's like that Easter egg hunt all over again. The x-intercepts are the values of x that make y zero. In other words, the solutions to the quadratic equation
0 = ax2 + bx + c.
This may involve using the quadratic formula. Since not all quadratic equations have solutions, the graph might not have any x-intercepts. It could look like this, for example:
The y-intercept is the value we get when we plug x = 0 in to the equation ax2 + bx + c. When we do that, we find
a(0)2 + b(0) + c = c.
So the y-intercept is just y = c. A parabola will always have a y-intercept, since c will always be some number (possibly 0). With all of this up-in-the-air x-intercept business, it's nice to know we can still rely on c and the y-intercept. They'll stay with us through thick and thin.
2. Finding the vertex.
When a parabola has x-intercepts, the vertex shows up halfway between them. The number halfway between two numbers is also known as their average. Just like how you're halfway between the ages of your older brother and younger sister, which makes you totally average. Wait...
The average of 4 and 10 is , which is the number halfway between 4 and 10. If we use the quadratic formula to find the x-intercepts, we get the values
The number halfway between these two values is
This means the vertex occurs at .
But wait, what happens when the quadratic formula doesn't get us any solutions? Sneakily enough, the vertex would still be at . The value will always exist, because if a = 0 we don't have a quadratic formula in the first place. Oh, algebra. Always trying to slip one past us. Not this time, pal.
3. Deciding whether the graph opens upwards ("U'') or downwards (upside-down "U'').
The coefficient a on the x2 term tells us whether the graph opens upward or downward. If a is positive, the graph opens upward. If a is negative, the graph opens downward. This should be easy to remember, because when you have a positive attitude your mouth forms a "U" shape, but when you're being negative it forms an upside-down "U" shape. Unless you're one of those people who can remain completely straight-faced at all times, in which case you're on your own with this one.
Anyway, the reason for this is that as x gets farther from 0, y will be getting more and more negative.
The exponential functions we'll deal with here are functions of the form
y = ab(linear function of x) + c
where a and c are real numbers, and b is greater than 1. Really, this just means we have a number greater than 1 getting raised to the x. Numbers less than 1, you can catch the next train to Outtahereville.
The simplest kind of exponential function would be something like:
y = 2x
Instead of just raising to the x, we could also raise 2 to a linear function of x, such as:
y = 2(0.5x + 1)
We could multiply by something:
y = 4(2)(0.5x + 1)
We could also add a constant:
y = 4(2)(0.5x + 1) + 11
All these things are exponential functions. Some are uglier than others, but luckily for them, we find beauty within.
How does this translate to graphs?
Graph the function y = 2x.
If we plug in x = 0 we get y = 20 = 1, so that gives us the point (0, 1), which takes care of the y-intercept.
This function won't have any x-intercepts, since no value of x will satisfy the equation 2x = 0.
Ugh, so hard to please.
For the general shape of the graph, let's find a few more points and see what happens.
Okay, so as x goes to the right, y gets bigger (and does so quickly). What happens as x goes to the left?
Hm. As x goes to the left, the values of y get really tiny. Maybe they just stole a shrink ray from a top-secret government lab. They'll never hit zero, so this function has no x-intercepts, but they get very, very close. So close that when we try to draw the graph, it'll look like the function is hitting the x-axis. It's like when your little brother used to put his hand real close to your face without actually brushing your skin and taunt you by shouting, "I'm not touching you!"
Now we connect the dots in a nice, curvy shape:
This picture shows the general shape of an exponential function. All exponential functions will look like this. They may be turned upside-down or shifted around, but they'll all have roughly this same kind of curve. Any exponential function will also have an asymptote—a value that the function gets really close to, but never quite hits. In our previous example with your little brother, your face would be the asymptote.
The asymptote for the function y = 2x is the line y = 0. We can draw an asymptote in a graph by drawing a dashed line:
The dashed line indicates that the function never quite reaches that value. It also indicates that vehicles are allowed to pass freely across it, as long as they check first .
Graph the function y = 2(0.5x + 1).
This function still won't have any x-intercepts, since 2(0.5x + 1) is not 0 for any value(s) of x. Instead of being 1 when x = 0, this function will be 1 when x = -2, since then we'll get:
2(0.5(-2) + 1) = 20 = 1
Let's keep point-hunting.
This graph looks very similar to the previous graph. Separated at birth much?
Graph the function y = 2-x.
Let's find some points:
We seem to have turned the graph around. Next up, we'll be attempting to turn the beat around.
As x gets bigger, y is now getting smaller. As x gets smaller and heads off the page to the left, y is getting bigger. The graph looks like this:
A fancy way to say this is that the function y = 2-x is what we get when we reflect y = 2x across the y-axis.
Graph the function y = -3(2)x.
What will happen now? Let's think about it before we find points. Maybe we can pull a Nostradamus and predict this sucker.
For starters, all the y-values of the function will be negative, as there's still no way to get 0. When x is positive, 2x will be "far away'' from 0, so -3(2)x will also be far away from 0. It's okay, they can still write or send a postcard. If we find a couple of points, we see this is exactly what happens:
How about when x is negative? In that case, 2x will be "close" to 0, so -3(2)x will also be close to 0. It's not surprising. After all, they did grow up together.
If we find some points with negative values of x, we see that as x becomes smaller (more negative), y becomes closer to 0:
If we graph all the points we have so far and connect the dots with a curve, we find the same general shape as the graph of y = 2x, only upside-down:
Let's review some of the things we've figured out from the examples so far. In general, adding a constant to the end of a function moves the graph of that function up or down. Hopefully no bratty kid makes his way into the function, or he may hit all the buttons so the graph stops at every floor.
The graph of y = 3x + 5 will be 5 higher than the graph of y = 3x, and the graph of y = 3x – 5 will be 5 lower than the graph of y = 3x.
The asymptote of an exponential function is given by the constant term. As we can see from the graph, the asymptote of y = 3x + 5 is y = 5, while the asymptote of y = 3x – 5 is y = -5.
A function such as y = 3x, when graphed, makes a nice curve:
If we switch the sign on the exponent, the graph turns around, as if startled:
And if instead of switching the sign on the exponent we multiply the whole function by -1, the graph turns upside-down, as if incredibly startled:
One big question at this point is, "Why should I care? The calculator can draw it for me. I bow to and worship the electronic mini-god." Well, okay, but sometimes, particularly on an exam, it will save valuable time to be able to look at a graph like:
...and recognize it as an exponential graph where the exponent is negative. Because it'll be saving you time, and because time is money, being able to recognize an exponential graph makes you money. Or something like that.
An inequality is just a type of relation, which means we can graph it like we would graph any relation. The trick with inequalities is that, instead of drawing lines to connect the dots, we have to shade in big areas of the graph. You could use some shade though. You're starting to burn.
Graph the inequality y ≥ x.
Let's find some points in the relation. This relation will contain every ordered pair of real numbers in which y is at least as big as x.
We can plot the points we have so far:
We can see that every point along the line y = x will be included in the relation, since for any point on that line y is at least as big as x. Good news for all you line-lovers out there. Let's include this line in our picture:
We have some points already, which are all above the line. If we were to find more points between the ones we already have, we would find that they're all in the relation. The relation contains every point in or above the line y = x, which we show by shading in that half of the graph. You might need to take an extra trip to Staples. You're really going to start burning through your pencil supply.
A linear inequality is an inequality that can be written with y on one side and a linear polynomial in x on the other. Let's look at these for a bit, since they're easier to think about that other inequalities. We promise we'll try to make your brain hurt more later. To graph a linear inequality, we
We don't generally advocate taking sides, but in this instance it's perfectly acceptable.
Graph the inequality y + 3 ≤ 2x.
The first thing we do is graph the line
y + 3 = 2x
which is also known as the line
y = 2x – 3.
Then we need to figure out which side of the line we want to be on. We can do that by picking a point on one side of the line, and seeing whether or not that point satisfies the inequality, i.e. whether that point is in the relation. Let's pick an easy point, like (1, 4). Sorry to refer to him as easy, but it's true. He's an undeniably gullible pushover.
When x = 1 and y = 4, our original inequality becomes:
4 + 3 ≤ 2(1)
7 ≤ 2
That's definitely not true—7 sure isn't smaller than 2—so this point does not satisfy the inequality. That means we shade in the other side of the graph:
When an inequality is strict (like, with a "<" or ">" symbol instead of "≤" or "≥"), we do the same thing as above except that now we don't want to include the points on the line. In fact, we don't want to do anything that might upset it, considering how strict it is. We are so over being grounded.
To show that we aren't including the points on the line, we draw a dashed line instead of a solid line. For example, if we wanted to graph y + 3 < 2x instead of y + 3 ≤ 2x, we'd use the exact same graph as above, except we'd draw a dashed line and shade everything underneath it.