Answer: y = -3x^2 + 24x - 45
Explanation:
To find the equation of the parabola with a vertex of (4, 3) and passing through points (2, -9) and (5, 0), we can use the vertex form of a parabolic equation:
y = a(x - h)^2 + k
Here, (h, k) is the vertex of the parabola. So, we have:
y = a(x - 4)^2 + 3
Now, we need to find the value of 'a' using the two points (2, -9) and (5, 0).
First, let's plug in the point (2, -9):
-9 = a(2 - 4)^2 + 3
-9 = a(-2)^2 + 3
-9 = 4a + 3
Subtract 3 from both sides:
-12 = 4a
Now, divide both sides by 4:
a = -3
Now that we have the value of 'a', we can plug it back into the equation:
y = -3(x - 4)^2 + 3
This is the equation of the parabola in vertex form. To convert it to standard form, we can expand the equation:
y = -3(x^2 - 8x + 16) + 3
y = -3x^2 + 24x - 48 + 3
Now, combine the constants:
y = -3x^2 + 24x - 45
This is the equation of the parabola in standard form.