Question

write the equation in vertex form for the function

(-3,-4) and (-2,3)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the vertex form of the equation for the function given the points (-3, -4) and (-2, 3), we need to follow these steps:

Step 1: Find the axis of symmetry (x-value of the vertex):
The axis of symmetry is the average of the x-coordinates of the given points:
x = (-3 + (-2)) / 2 = (-5) / 2 = -2.5

Step 2: Find the y-coordinate of the vertex:
Substitute the x-value of the axis of symmetry back into the function to find the y-coordinate:
y(-2.5) = k
Solve for k using one of the points, let's use (-3, -4):
-4 = a(-3 + 2.5)^2 + k
-4 = a(-0.5)^2 + k
-4 = a(0.25) + k
-4 = 0.25a + k

We also know that y(-2) = 0:
0 = a(-2 + 2.5)^2 + k
0 = a(-0.5)^2 + k
0 = a(0.25) + k
0 = 0.25a + k

Now, we can solve the system of equations to find the value of k:
-4 = 0.25a + k
0 = 0.25a + k

Subtract the second equation from the first equation:
-4 = 0.25a + k
0 = -0.25a - k
-4 = 0.5k
k = -4

Step 3: Substitute the value of k back in and find the value of a:
-4 = 0.25a - 4
0 = 0.25a - 4
0 = 0.25a - 4
4 = 0.25a
a = 16

Step 4: Write the equation in vertex form:
f(x) = 16(x + 2.5)^2 - 4

Therefore, the equation in vertex form for the function given the points (-3, -4) and (-2, 3) is f(x) = 16(x + 2.5)^2 - 4.