a parabola has an axis of symmetry at x=-7, a maximum height of 4 and also passes through the point (-6,0). Write the equation of the parabola in vertex form.
This is the deal:
Since the axis of symmetry is a vertical line, x = -7 and the parabola has a maximum height of 4 units above the x axis; we can safely conclude that it opens down and its vertex is at (-7, 4). Besides, it passes through the point (-6, 0), which is below the vertex; definitively, it opens down. Furthermore, if it would open up, it would have a minimum height, not a maximum; so, no arguments, it opens down.
The vertex form of the equation is y = a(x - h)2 + k
Since the vertex is defined as (h, k), h = -7 and k = 4:
y = a(x - (-7))2 + 4 or: y = a(x + 7)2 + 4
We just need to find the value of "a" (which must be negative, remember, it opens down). Here is where we use the point through which the parabola passes through: (-6, 0).
We substitute the values x = -6, y = 0 in the equation and solve for a:
0 = a(-6 + 7)2 + 4 simplify and subtract 4 on both sides of the equation:
-4 = a(1)2 or a = -4 Now we are ready to write our completed equation:
y = -4(x + 7)2 + 4 A lot of fun, right?