a parabola has vertex (-6,-12) AND PASSES THROUGH (6,24) find its equation in the form ax^2+bx+c
let the equation be in the form
y = a(x-h)^2 + k, where (h,k) is the vertex
so we have
y = a(x+6)^2 - 12
plug in the point (6,24) to get a
24 = a(12)^2 - 12
a = 36/144 = 1/4
so y = (1/4)(x-6)^2 - 12
expand this to change it into the required form
Please check my arithmetic, I am still on my first cup of coffee.
To find the equation of a parabola in the form ax^2 + bx + c, we need to determine the values of a, b, and c.
1. Start with the general form of a parabola equation: y = ax^2 + bx + c.
2. We know that the vertex of the parabola is given by the coordinates (-6, -12). Since the vertex lies on the parabola, we can substitute these values into the equation to get: -12 = a(-6)^2 + b(-6) + c.
3. Now, we need additional information to find the specific values of a, b, and c. We are given that the parabola passes through the point (6, 24). Substitute these values into the equation: 24 = a(6)^2 + b(6) + c.
4. We now have a system of two linear equations with three unknowns (a, b, c). To solve for the values, we can use a method called substitution or elimination.
Substitution method:
- From equation 2, isolate c: c = -12 - a(-6)^2 - b(-6).
- Substitute this value of c into equation 1: 24 = a(6)^2 + b(6) + (-12 - a(-6)^2 - b(-6)).
- Simplify the equation and solve for a and b.
Elimination method:
- Multiply equation 2 by -1 to make the coefficients of c opposite in signs: -24 = -a(6)^2 - b(6) - c.
- Add this equation to equation 1 to eliminate the variable c: -12 + (-24) = a(-6)^2 + b(-6) - a(6)^2 - b(6).
- Simplify the equation and solve for a and b.
5. After finding the values of a and b, substitute them back into one of the original equations to find the value of c.
6. Finally, write the equation of the parabola in the form ax^2 + bx + c using the values of a, b, and c that you found.