Question

Write a function in vertex form if the vertex is (2,4), and a non-vertex point is (0,-16).

Answer 1

To write the equation of a parabola in vertex form, we use the formula:

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

where \((h, k)\) is the vertex of the parabola. In this case, the vertex is given as \((2, 4)\), so we can substitute \(h = 2\) and \(k = 4\) into the equation:

\[
y = a(x - 2)^2 + 4
\]

Next, we need to find the value of \(a\). We can use the non-vertex point \((0, -16)\) to solve for \(a\). Substitute \(x = 0\) and \(y = -16\) into the equation:

\[
-16 = a(0 - 2)^2 + 4
\]

This simplifies to:

\[
-16 = a(4) + 4
\]

\[
-16 - 4 = 4a
\]

\[
-20 = 4a
\]

\[
a = -5
\]

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

\[
y = -5(x - 2)^2 + 4
\]

Thus, the function in vertex form is:

\[
\boxed{y = -5(x - 2)^2 + 4}
\]