Question

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

[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.]

Vertex left parenthesis, 2, comma, –, 1, right parenthesis(2,–1) and point on the parabola left parenthesis, 4, comma, 3, right parenthesis(4,3)

Answer 1

To find the equation of the parabola in vertex form, we can use the given vertex and a point on the parabola. The vertex form of a parabola is given by:

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

where \((h, k)\) is the vertex of the parabola and \(a\) is a coefficient that determines the width and direction of the parabola.

Given:
- Vertex: \((h, k) = (2, -1)\)
- Point on the parabola: \((4, 3)\)

Substituting the vertex into the vertex form gives us:

\[
f(x) = a(x - 2)^2 - 1
\]

Next, we need to find the value of \(a\). We can do this by substituting the point \((4, 3)\) into the equation:

Substituting \(x = 4\) and \(f(x) = 3\):

\[
3 = a(4 - 2)^2 - 1
\]

Simplifying:

\[
3 = a(2)^2 - 1
\]
\[
3 = 4a - 1
\]

Now, add 1 to both sides:

\[
4 = 4a
\]

Now, divide both sides by 4:

\[
a = 1
\]

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

\[
f(x) = 1(x - 2)^2 - 1
\]

This simplifies to:

\[
f(x) = (x - 2)^2 - 1
\]

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

\[
f(x) = (x - 2)^2 - 1
\]