Question 203258
Given:
{{{ax^2+4x+c}}}
Find a and c.
First, you are told that the maximum (vertex) of the parabola is at (1, 8).
The value of the x-coordinate of the vertex is given by:
{{{x = (-b)/2a}}} Substitute b = 4 and x = 1 (from (1, 8)) to get:
{{{1 = -4/2a}}} Multiply both sides by a.
{{{a = -4/2}}} Simplify.
{{{highlight(a = -2)}}} so we have...
{{{y = -2x^2+4x+c}}} Now to find the value of c, just substitute the x- and y-coordinates of any of the points given in the problem. (1, 8) or (-1, 0) or (0, 6) or (3, 0).
The reason you can do this is because every one of these points lies on the curve (parabola) and thus satisfies the given equation.
Let's choose the point (0, 6).
{{{6 = -2(0)^2+4(0)+c}}} Simplify.
{{{highlight_green(6 = c)}}}
The equation is:
{{{highlight(y = -2x^2+4x+6)}}}
Let's see what the curve looks like!
{{{graph(400,400,-5,5,-5,10,-2x^2+4x+6)}}}