Answer:
Three equations I found with two variables:
1. -20 = a(2 - h)^2 + k
2. 0 = a(-2 - h)^2 + k
3. 0 = a(3 - h)^2 + k
Explanation:
To write the equation of a parabola with x-intercepts (-2,0) and (3,0) that passes through the point (2,-20), you can use the standard form of a quadratic equation:
y = a(x - h)^2 + k
In this equation:
- (h, k) is the vertex of the parabola.
- (x, y) is any point on the parabola.
- 'a' is a constant that determines the direction and width of the parabola.
Since the parabola has x-intercepts at (-2,0) and (3,0), we know that its roots are -2 and 3. This means that (x + 2) and (x - 3) are its factors. To find the equation, we can plug in the point (2,-20) to solve for 'a':
-20 = a(2 - h)^2 + k
Now, plug in the x-coordinate and y-coordinate of the given point (2,-20):
-20 = a(2 - h)^2 + k
-20 = a(2 - h)^2 - 20
Next, we use the x-intercepts to find h and k:
- When x = -2 (one x-intercept), y = 0, so:
0 = a(-2 - h)^2 + k
- When x = 3 (the other x-intercept), y = 0, so:
0 = a(3 - h)^2 + k
Now, we have a system of three equations with two variables (a, and h):
1. -20 = a(2 - h)^2 + k
2. 0 = a(-2 - h)^2 + k
3. 0 = a(3 - h)^2 + k