Intro
The formula for the vertex form of a quadratic equation is y=a(x-h)^2+k. In this formula, h=x and k=y when it comes to finding the vertex. We use this formula to graph parabolas.
USING VERTEX FORM TO GRAPH A PARABOLAGraphing a parabola is easy in vertex form. Below is an example of a problem. Here are the steps to graph...
1. Graph the vertex and the axis of symmetry. Below, the vertex is (2,5) because in the equation h=2 and k=5. The axis of symmetry is whatever x equals. This will always be vertical. Draw a dotted line to keep your place. 2. Find and graph another point. Plug in any number (try 0) into x in your equation then solve. Your answer will be the y and the number you plugged in will be the x in your point. I plugged in 0 and ended up with (0,13). 3. Graph the corresponding point to the point you just found. My point was (0,13) so I have to find the point that mirrors it on the other side of the axis of symmetry. That point will have a different x but the same y. My corresponding point ended up being (4,13). 4. Sketch the curve and you're done! |
Writing the equation of a parabola
Writing the equation of a parabola is also easy with vertex form. Look at the example parabola below.
From the parabola we can see the vertex, axis of symmetry (although it is not dotted here) and one other point. Now all we need to do is write the equation.
1. Start off with vertex form. y=a(x-h)^2+k
2. Remember that h=x and k=y when we find the vertex. The vertex above is (3,4) so plug that into the equation. y=a(x-3)^2+4
3. The other point they gave us is (5,-4) so plug those into the x and y spots in our equation. -4=a(5-3)^2+4
4. Simplify that problem. We end up with -4=4a
5. Finally, divide and solve for a. -2=a
The final equation is y=-2(x-3)^2+4
1. Start off with vertex form. y=a(x-h)^2+k
2. Remember that h=x and k=y when we find the vertex. The vertex above is (3,4) so plug that into the equation. y=a(x-3)^2+4
3. The other point they gave us is (5,-4) so plug those into the x and y spots in our equation. -4=a(5-3)^2+4
4. Simplify that problem. We end up with -4=4a
5. Finally, divide and solve for a. -2=a
The final equation is y=-2(x-3)^2+4
real world connection
Hopefully you noticed the roller coaster picture at the top of the page. She might've been on a roller coaster with loops and drops. Roller coasters are an example of real life parabolas. Here are a couple of real world examples...
This is a picture of the Medusa ride at Six Flags Great Adventure in New Jersey. It is the world's first floorless roller coaster. The curve can be modeled with the function y=0.0001432(x-2130)^2. where x and y are measured in feet. The origin of the function's graph is at the top of each drop. How far apart are the drops? How high are they?
Start by drawing a diagram. The function is in vertex form. Since h=2130 and k=0, the vertex is at (2130,0). The vertex is halfway between the drops, so the distance between the drops is 2(2130ft)=4260 ft. To find the tower's height, find y for x=0. y=0.0001432(0-2130)^2 y=0.0001432(-2130)^2 ≈650 The drops are 4260 ft apart and about 650 ft high. |