Sure! Let's find the value of [tex]\( a \)[/tex] and write the equation of the parabola step by step.
1. Identify the vertex and the formula of the parabola:
- The vertex is [tex]\((-2, 5)\)[/tex].
- The formula for a parabola in vertex form is:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. In our case, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-2, 5)\)[/tex].
2. Substitute the vertex coordinates into the formula:
[tex]\[
y = a(x + 2)^2 + 5
\][/tex]
3. Substitute the given point [tex]\((-3, 2)\)[/tex] into the formula to find [tex]\( a \)[/tex]:
- The point [tex]\((-3, 2)\)[/tex] lies on the graph of the parabola. This means when [tex]\( x = -3 \)[/tex], [tex]\( y = 2 \)[/tex].
[tex]\[
2 = a(-3 + 2)^2 + 5
\][/tex]
[tex]\[
2 = a(-1)^2 + 5
\][/tex]
[tex]\[
2 = a(1) + 5
\][/tex]
[tex]\[
2 = a + 5
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
[tex]\[
2 - 5 = a
\][/tex]
[tex]\[
a = -3
\][/tex]
5. Write the equation of the parabola:
Now that we have found [tex]\( a = -3 \)[/tex], we can substitute it back into the formula:
[tex]\[
y = -3(x + 2)^2 + 5
\][/tex]
So, the value of [tex]\( a \)[/tex] is [tex]\( -3 \)[/tex] and the equation of the parabola is:
[tex]\[
y = -3(x + 2)^2 + 5
\][/tex]
