Given the following graph of a parabola, find its equation, in the form
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
We can see that the vertex of the parabola is at point (2,1). This means that ℎ=2 and 𝑘=1.
Next, we can choose another point on the parabola, for example (0,-3) or (4,-3), and substitute these values into the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 to solve for 𝑎.
Using (0,-3), we have:
-3 = 𝑎(0 − 2)2 + 1
-3 = 4𝑎 + 1
-4 = 4𝑎
𝑎 = -1
Therefore, the equation of the parabola is:
𝑦 = -1(𝑥 − 2)2 + 1
or
𝑦 = -(𝑥^2 - 4𝑥 + 4) + 1
or
𝑦 = -𝑥^2 + 4𝑥 - 3
To find the equation of a parabola in the form 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘, you need to determine the values of 𝑎, ℎ, and 𝑘 from the given graph.
Step 1: Identify the vertex of the parabola.
The vertex is the highest or lowest point on the parabola and is given by the coordinates (ℎ, 𝑘).
Step 2: Determine the value of 𝑎.
The value of 𝑎 determines the shape and direction of the parabola. If 𝑎 is positive, the parabola opens upwards, and if 𝑎 is negative, the parabola opens downwards.
Step 3: Write down the equation using the values of ℎ, 𝑘, and 𝑎.
Substitute the values of ℎ, 𝑘, and 𝑎 into the equation 𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘.
If you provide me with the specific details or coordinates of the vertex and another point on the parabola, I can guide you through the process of finding the equation step-by-step.