. Given the following graph of a parabola, find its equation, in the form
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
Without the exact coordinates of the points on the graph, it is not possible to determine the equation of the parabola. More information is needed.
To find the equation of a parabola in the form 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘, you will need to identify three key points on the graph. These points are the vertex (ℎ, 𝑘), and any two other points on the parabola. Once you have these points, you can plug them into the equation to solve for the values of 𝑎, ℎ, and 𝑘.
Let's say the vertex is at point (3, 2). Now, find two other points on the parabola. It is always easiest to choose points that are symmetrically placed on either side of the vertex. Let's choose (2, 3) and (4, 3) as our other points.
Next, substitute the coordinates of each point into the equation 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘. After substituting, you will have three equations:
1. For vertex (3, 2):
2 = 𝑎(3 − ℎ)² + 𝑘
2. For point (2, 3):
3 = 𝑎(2 − ℎ)² + 𝑘
3. For point (4, 3):
3 = 𝑎(4 − ℎ)² + 𝑘
Now, you can solve this system of equations to find the values of 𝑎, ℎ, and 𝑘. One way to do this is by first subtracting the second equation from the first equation, and then subtracting the third equation from the first equation. This will eliminate the 𝑘 term:
2 - 3 = 𝑎(3 - ℎ)² - 𝑎(2 - ℎ)²
1 = 𝑎(3 - ℎ)² - 𝑎(2 - ℎ)²
2 - 3 = 𝑎(3 - ℎ)² - 𝑎(4 - ℎ)²
-1 = 𝑎(3 - ℎ)² - 𝑎(4 - ℎ)²
Simplify these equations:
-1 = -𝑎(ℎ² - 6ℎ + 9) + 𝑎(ℎ² - 8ℎ + 16)
-1 = -𝑎ℎ² + 6𝑎ℎ - 9𝑎 + 𝑎ℎ² - 8𝑎ℎ + 16𝑎
-1 = -2𝑎ℎ + 7𝑎 + 16𝑎 - 9𝑎 - 8𝑎ℎ + 6𝑎ℎ
Combine like terms:
-1 = 4𝑎ℎ - 𝑎
0 = 4𝑎ℎ - 𝑎 + 1
Now, you can solve this equation to find the value of 𝑎. Once you have 𝑎, you can substitute it back into any of the three original equations to solve for ℎ and 𝑘.
After finding the values of 𝑎, ℎ, and 𝑘, you can write the equation of the parabola in the form 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘.