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Math 101: College Algebra11 chapters | 81 lessons
Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.
A big part of a college algebra class is getting introduced to the different types of relationships we see in math. The most basic is a linear function, which only has plain xs (such as y = 2x + 4). But once you get past those, the next step is to a quadratic function , which has xˆ2s (such as y = x^2 + 4). There's a lot to learn about quadratics, but the best place to start is with their graphs.
Anytime you graph a quadratic equation you end up with what is called a parabola. Parabolas have been behind the scenes of sports, celebrations, and wars for ages. When the first javelin was thrown in the Greek Olympics, or when the first firework was launched in China, or even when the first cannon was fired in the Civil War, they all flew through the air in the shape of a parabola.
Today, parabolas are still around in things just like this, but they've also made their way into more modern inventions, like video games. Back around 2007, I actually had an idea for a video game that would use parabolas. I thought it might be fun to just shoot things across the screen. So, what if these birds had their eggs stolen by some pigs, and the birds got really angry and wanted to get back at the pigs to get their eggs back. Maybe we could make these birds shoot across the screen with slingshots, and see if we could attack the pigs and get our eggs back.
So, here we've got a likely scenario. Let's try to launch our bird that's really angry at that mean old pig, and let it fly. Eh, we missed. Let's try again. I want to aim up a little bit more this time, let's try again. And, yeah! We got him.
The game draws in those little dots to help you aim your shots, but the path they sketch out is actually a perfect parabola. Notice that depending on the angle we launch the bird at, we get a slightly different shape. But, even if we shot the bird almost straight up, or even really close to the ground, it would still be a parabola because there are lots of different kinds.
Depending on how we shoot the bird, each parabola would have a different maximum height, which is our first vocabulary word. The maximum is the highest y-value that the parabola reaches. In this case, that represents the height that the bird gets.
The name of the actual point on the parabola where it gets to the maximum is our second vocab word; it's called the vertex. You might say that the vertex is in the middle of the parabola. That's because the parabolas are symmetrical, they're the same on either side. This means our third vocab word is the line that goes straight down through the middle of the parabola to divide it in half. It's called, the axis of symmetry.
Just like any other graph, parabolas' intercepts where the curve intersects either the x or the y-axis. In parabola word problems, the x-intercepts will often be the place where your flying object hits the ground, just like here. These x-intercepts of quadratic equations (and also bigger functions) can also be called roots.
For our last few vocabulary terms, we'll need to abandon our video game analogy, or maybe, imagine a new version on some crazy backwards planet where gravity is upside down. This is because parabolas can be concave down like the examples we've been talking about, or concave up, which means the whole shape is just flipped upside down. All the vocab we've talked about is exactly the same for concave up parabolas except one. Now, instead of our vertices being a maximum, they indicate the minimum that the parabola will reach.
To review, parabolas are the shape that graphs of quadratic equations take. They look kind of like a big letter U, and happen anytime something is launched into the air. They can be concave up or concave down, have vertices where a maximum or minimum happens, intercepts where they cross one of the two axes and an axis of symmetry that divides them in half.
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