Back To CourseAlgebra II: High School
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
In this video lesson, we are going to be talking about parabolas. What is a parabola? We can call it an arc. Parabolas are actually really useful shapes that we use for very important applications in the world. You see them in use in those satellite dishes for watching television. Satellite dishes are also used to communicate with the machines we have in outer space. Because of all these very useful applications, learning how to graph parabolas from equations is a very useful skill to have.
One interesting fact that is important for you to know is that every parabola has a point called the focus, where the distance from a point on the parabola to the focus is the same as the distance from that point to a straight line underneath the parabola. Since we are working in algebra, of course we have an equation for our parabola.
When we graph our parabolas, we can draw them curving either up and down or curving to the left and right. Because of these two different directions, we have two different equations for each. We call the parabolas that curve up and down vertical parabolas, and we call the parabolas that curve sideways horizontal parabolas.
The standard form of a vertical parabola is (x - h)2 = 4p(y - k), where h, k and p are numbers that give you the location of the focus. For the vertical parabola, the focus is given by the point (h, k + p). For example, the parabola (x - 3)2 = 8(y - 2) has a focus of (3, 4) since our h equals 3, our p equals 2 and our k equals 2. How come our p equals 2? Well, the standard form has a 4p outside the parentheses with the y, and our equation has an 8 in its place. So, what multiplied by 4 will get me to 8? It's a 2.
The standard form of a horizontal parabola is (y - k)2 = 4p(x - h), where the focus is given by the point (h + p, k). For example, the parabola (y - 4)2 = 4(x - 1) has a focus of (2, 4) since our h equals 1, our p equals 1 and our k equals 4. Our p equals 1 since we have a 4 in the 4p place, and 4 times 1 will equal 4.
Notice how the k is always linked with the y, and the h is always linked with the x. The p is always linked with the part that is not squared.
Because these standard form equations allow us to easily find the focus point, they are also very useful for us when we want to graph our parabolas. Why is this? Because once we find our focus point, we can use the location of the focus point to find the location of the tip of our parabola. We can then find points on either side of the tip so we can graph our parabola. Let's go ahead and see how we can do this.
Why don't we go ahead and graph the vertical parabola (x - 3)2 = 8(y - 2). We already found the focus of (3, 4). The way we are going to find the location of the tip of our parabola is to take the x value of our focus and plug it into our parabola equation to find the y value of the tip. We are using the x value because our parabolas are facing up and down, so our arc dips up and down from left to right. If we plug in our x value of 3, we get (3 - 3)2 = 8(y - 2), which becomes 0 = 8(y - 2). To solve for y, we divide by 8 on both sides, and then we add the 2. Doing this we get 0 = y - 2, which becomes y = 2. So our tip of our parabola is (3, 2).
Now we can go ahead and plot two points on either side of our tip to see how our parabola curves. We will plug in x = 2, x = 1, x = 4 and x = 5 to see how our parabola curves on either side of the tip. Each time, we will plug in our x, and then solve for y. We can build a table of our points to plot once we get to the actual graphing part.
Notice how the points on either side of the tip match each other in their y value? That tells you that you are doing something right. Now that we have our points, we can go ahead and graph them, and then draw out our arc.
Notice how the points show you how the curve begins. All you have to do is to follow this curve and you will get your parabola.
We just graphed a vertical parabola. What about graphing a horizontal parabola? Let's try one of these as well. We will graph the parabola (y - 4)2 = 4(x - 1). We already found the focus of (2, 4). Here, instead of plugging in for x to find our y, we will plug in our y to find our x. So to find our tip, we will plug in our y value from our focus to find the x value of our tip. Plugging in our y value of 4 we get (4 - 4)2 = 4(x - 1). Solving this for x gives x = 1. So the tip of our parabola is located at (1, 4).
Remember now that our parabola opens to the side so instead of going to either side of our tip using the x value, we will now go on either side of our tip using the y value. So, we will plug in two y values above our tip and two y values below our tip. We will find our x for y = 3, y = 2, y = 5 and y = 6. We will do the same as before and build a table.
Now we can go ahead and plot the points, and then draw out the curve like we did before. Just remember that now our curve is sideways, or horizontal.
Now, let's review what we've learned. We've learned that a parabola looks like an arc. Satellite dishes are real life examples of parabolas. One interesting fact about parabolas is that they have a point called the focus, where the distance from a point on the parabola to the focus is the same as the distance from that point to a straight line underneath the parabola.
We learned that there are two different forms of the standard form equation for parabolas. The standard form of a vertical parabola, a parabola opening either up or down, is (x - h)2 = 4p(y - k), where the focus is (h, k + p). The standard form of a horizontal parabola, a parabola opening either to the left or right, is (y - k)2 = 4p(x - h), where the focus is (h + p, k).
To graph out a vertical parabola, we plug in the x value of our focus point into our parabola equation to find the y value of the tip of our parabola. We then choose two x values on either side of the tip of our parabola to see how the parabola curves. We find the y values for these x values and then we plot the points, and then draw our parabola. To graph a horizontal parabola, we plug in the y value of our focus point into our parabola equation to find the x value of the tip of our parabola. Then we choose two y values on either side of the tip. We evaluate our parabola equation to find the x value at these y values. We then plot the points and draw our parabola.
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Back To CourseAlgebra II: High School
23 chapters | 203 lessons