Question

A graph.Short description, A graph.,Long description,
On a coordinate plane, a parabola opens up. It has a vertex at (negative 2, negative 5), crosses the y-axis at (0, 3), and goes through the point (negative 4, 3).


Question
Which is the equation of the parabola?

Answer options with 4 options
A.
y equals one-half left-parenthesis x minus 2 right-parenthesis squared minus 5

B.
y equals one-half left-parenthesis x plus 2 right-parenthesis squared minus 5

C.
y equals 2 left-parenthesis x minus 2 right-parenthesis squared minus 5

D.
y equals 2 left-parenthesis x plus 2 right-parenthesis squared minus 5

Answer 1

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

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

where \((h, k)\) is the vertex of the parabola.

Given:
- Vertex: \((-2, -5)\) (which means \(h = -2\) and \(k = -5\))

Thus, the equation initially looks like:

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

Next, we'll use the point on the parabola \((0, 3)\) to find the value of \(a\).

Substituting \((x, y) = (0, 3)\):

\[
3 = a(0 + 2)^2 - 5
\]

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

\[
3 = 4a - 5
\]

Adding 5 to both sides gives:

\[
8 = 4a
\]

Dividing by 4 gives:

\[
a = 2
\]

Now, substituting back, we get:

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

This corresponds to option D:

**D. \(y = 2(x + 2)^2 - 5\)**

To verify, we can check if the point \((-4, 3)\) lies on this equation:

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

\[
y = 2(-2)^2 - 5
\]

\[
y = 2(4) - 5
\]

\[
y = 8 - 5 = 3
\]

The point \((-4, 3)\) indeed lies on the parabola, confirming our calculation.

Final answer:

**D. \(y = 2(x + 2)^2 - 5\)**