The Equation
Just like with circles, an ellipse has a center. Unlike circles, an ellipse has two different measurements for its radius, which is the distance from the edge to the center. Notice how an ellipse is longer in one direction than the other? Because of this, an ellipse will have a radius measurement for one direction and another radius measurement for the other direction. Our equation of an ellipse takes into account all of these measurements. Our equation is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center of our ellipse, a is the radius in the horizontal direction, and b is the radius in the vertical direction. Notice that there is always a plus in between the x and y parts of the equation and that the equation always equals 1.
Graphing an Ellipse
Let's see how we can use this equation to help us graph an ellipse. Say we want to graph the equation x^2 / 4 + y^2 / 9 = 1. First, we notice that the center of our ellipse is located at (0, 0) since there are no numbers being subtracted or added to our x or y. Next, we figure out how far out our ellipse edges will go. Our a^2 is 4, so that means my a is 2. If my a is 2, then that means my ellipse goes out 2 spaces to the left and 2 spaces to the right of my center. I can go ahead and draw two dots at those two places. Next, I see that my b^2 is 9, which means my b is 3. Since my b is 3, then my other two dots will be drawn 3 spaces above my center and 3 spaces below my center. Now that I have my dots, I can go and draw my ellipse - my stretched-out circle - making sure to touch all those dots.
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Graphing another Ellipse
Let's try another one. Let's try graphing (x - 4)^2 / 9 + (y + 2)^2 / 4 = 1. The first thing I notice is that my center is at (4, -2) since my h is 4, and my k is -2. Why is my k a negative number? Notice how we have y + 2 in our problem. Our ellipse equation, though, shows y - k with a minus sign in between. Since we have y + 2, we are actually looking at y - (-2), so our k is -2. Next, we see that our a^2 is a 9, so that means a is 3. Our b^2 is a 4, so that means our b is a 2. Now we can go ahead and graph our four edge points. Since our a is 3, we plot a point 3 spaces to the left and another point 3 spaces to the right of our center. Our b is a 2, so we plot a point two spaces above and another point two spaces below our center. We then draw our ellipse and we are done!
Lesson Summary
Let's review now. An ellipse can be described as a stretched-out circle. Our ellipse equation is (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center of our ellipse, a is the radius in the horizontal direction and b is the radius in the vertical direction. To graph our ellipse, we find the center by looking for our h and k values. Our center will then be located at the point (h, k).
We then find our a and b values by taking the square roots of the denominators. For example, if the number located at the a^2 spot is 9, then my a is 3, since the square root of 9 is 3. Once we find the a and b values, we can plot the points that the edges of our ellipse will pass through. Finally, we graph our ellipse by drawing a stretched out circle that passes through all of our a and b points.
Learning Outcomes
Once you are done with this lesson you should be able to:
- Write the equation for an ellipse
- Identify the center and the horizontal and vertical radiuses of an ellipse from an equation
- Graph an ellipse
