Write the equation of the parabola in vertex form given the points below.

Question

[MAKE SURE to first solve for the a, minus, v, a, l, u, ea−value. Also, you must type f, of, x, equalsf(x)= for your equation.]

Maximum (Vertex) point left parenthesis, 6, comma, 5, right parenthesis(6,5) and a point on the parabola left parenthesis, 8, comma, minus, 3, right parenthesis(8,−3)

Answer 1

To find the equation of the parabola in vertex form, we start with the vertex form equation of a parabola, which is given by:

\[ f(x) = a(x - h)^2 + k \]

where \((h, k)\) is the vertex of the parabola. From the information given, the vertex \((h, k)\) is \((6, 5)\). Thus, our equation becomes:

\[ f(x) = a(x - 6)^2 + 5 \]

Next, we need to determine the value of \(a\). We can do this by using the second point given, which is \((8, -3)\). We will substitute \(x = 8\) and \(f(x) = -3\) into the equation to solve for \(a\):

\[ -3 = a(8 - 6)^2 + 5 \]

This simplifies to:

\[ -3 = a(2)^2 + 5 \] \[ -3 = 4a + 5 \]

Next, we isolate \(a\) by subtracting 5 from both sides:

\[ -3 - 5 = 4a \] \[ -8 = 4a \]

Now, divide both sides by 4:

\[ a = -2 \]

Now that we have \(a\), we can substitute it back into the vertex form equation:

\[ f(x) = -2(x - 6)^2 + 5 \]

Therefore, the equation of the parabola in vertex form is:

\[ f(x) = -2(x - 6)^2 + 5 \]