Today we're going to talk about parabolas. Parabolas are the name for the graphs that are created when you graph a quadratic equation. So we're going to look at some characteristics of parabolas, and we'll talk about how those characteristics can be seen in the quadratic equations associated with them.
So let's start by reviewing the characteristics of a parabola. I've got my first parabola graphed here. It's of the equation y is equal to x squared minus 4. I notice that it's an upward-facing U shape. And that's because the coefficient of the x squared term is positive, so it's going to be facing upwards. And we can see that our graph is approaching positive infinity on the y-axis, both as we go towards positive infinity on the x and towards negative infinity on the x.
The second parabola that I have graphed is of the equation y is equal to negative x squared plus 8x minus 16. So we see this as an upside-down U shape. And that's because the coefficient of the x squared term is negative. So this parabola is going towards negative infinity on the y-axis. And that's again both as we go towards positive or negative infinity on the x-axis.
So there are two important parts of parabolas that we want to also go over here. The first is the vertex, which is either the maximum or minimum point of our parabola. And the second and important part of a parabola is called the axis of symmetry. And the axis of symmetry is a vertical line that goes through the vertex of the parabola. So for this parabola, our axis of symmetry would be right on top of the y-axis, through x is equal to zero. And through this parabola, it would be through 4 on the x-axis.
And the axis of symmetry is a line of reflection, so any point, for example, that's on the right side of this axis of symmetry or the line of reflection can be reflected to create the points on the left side and vice versa. If we took the points on the left side, they could be reflected over the axis of symmetry to get the points on the right side.
So let's see how we can identify the vertex of a parabola or the graph of a quadratic equation, just by looking at its equation. So I've got the equation y is equal to 2x squared plus 4x plus 1, and I want to find the vertex. I have a formula for finding the x-coordinate of the vertex of a parabola, which is x is equal to negative b over 2a. So I can use that to find the x-coordinate of my vertex, and then I can substitute that value into my equation to find the y-coordinate.
In my formula, the values for b and a come from my equation. I know that a is the coefficient in front of the x squared term. b is the coefficient in front of my x term, and c is my constant term at the end. So looking at my formula, I have for my x-coordinate, that x will be equal to negative b-- so negative 4 over 2 times a, so 2 times 2. I can simplify this to be negative 4 over 4, which equals negative 1. So the x-coordinate of my vertex is negative 1.
And now I can find the y-coordinate of my vertex by substituting my value for x into my equation for the x, to find the value of y. So I'll have y is equal to 2 times negative 1 squared plus 4 times negative 1 plus 1. Simplifying, I have negative 1 squared, which will be positive 1. So this gives me 2 times 1.
Then I'm going to multiply. 2 times 1 will give me 1. 4 times negative 1 will give me negative 4 and then plus 1. And finally, adding from left to right, 2 plus negative 4 will give me negative 2, plus 1 will give me negative 1. So I found that the y-coordinate of my vertex is also negative 1. And so the vertex of my parabola written as a coordinate pair is negative 1, negative 1.
So let's go over vertex form of a quadratic equation. We call a quadratic equation written in this form vertex form, because we can simply look at the equation and the variables for h and k, to identify the vertex of the parabola. So the x-coordinate of my vertex is going to be the value that I have for my h variable. And the y-coordinate of my vertex is going to be the value that I have for my k variable.
So for example, this equation in vertex form tells me that my vertex is at negative 3, positive 2. And so you have to be careful, if you notice that the x-coordinate is negative 3, even though I have a plus sign. And that's because, to fit my formula for vertex form, I would have to be subtracting a negative 3, to have a positive here.
So when you're looking at your vertex, you need to make sure that you're careful with the sign of your h variable, so that it fits the formula. So a positive inside of my equation will give me a negative x-coordinate for my vertex.
So finally, let's look at another example of an equation that is written in vertex form and this corresponding graph, the parabola. So we know that in vertex form, we can quickly identify the vertex by looking at the values for h and k. So I see that my value for h will be 1. And my value for k will be negative 4.
So again, noting that because there's a minus in our formula, we ignore the minus and simply look at the numbers, so our value is positive. But because here we'd be adding a negative 4, our value for the y-coordinate is negative 4. So our vertex is at positive 1, negative 4, which matches the vertex that we see in our parabola.
So let's go over our key points from today. If the x squared coefficient in a quadratic equation is positive, the graph of the equation parabola will be pointing up. If the x squared coefficient is negative, the parabola will be pointing down. The axis of symmetry acts as a line of reflection, so all points on the left of the parabola can be reflected across the axis of symmetry to represent all points to the right and vice versa.
So I hope that these key points and examples helped you understand a little bit more about parabolas. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
a line of reflection passing through the vertex of a parabola; in up and down parabolas, it is a vertical line.
the maximum or minimum point of a parabola located on the axis of symmetry