Certainly! Let's find the function [tex]\( f(x) \)[/tex] whose graph is a parabola with the given vertex [tex]\((-1, 6)\)[/tex] and that passes through the point [tex]\((-2, -4)\)[/tex].
### Step-by-Step Solution
1. Vertex Form of Parabola:
The vertex form of a parabolic function is given by:
[tex]\[
f(x) = a(x - h)^2 + k
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Here, [tex]\((h, k) = (-1, 6)\)[/tex], so the equation becomes:
[tex]\[
f(x) = a(x + 1)^2 + 6
\][/tex]
2. Substitute the Point to Find [tex]\( a \)[/tex]:
The parabola passes through the point [tex]\((-2, -4)\)[/tex]. We substitute [tex]\( x = -2 \)[/tex] and [tex]\( f(x) = -4 \)[/tex] into the equation to determine the value of [tex]\( a \)[/tex].
Substituting, we get:
[tex]\[
-4 = a(-2 + 1)^2 + 6
\][/tex]
3. Solve for [tex]\( a \)[/tex]:
Simplify and solve for [tex]\( a \)[/tex]:
[tex]\[
-4 = a(-1)^2 + 6
\][/tex]
[tex]\[
-4 = a \cdot 1 + 6
\][/tex]
[tex]\[
-4 = a + 6
\][/tex]
Subtract 6 from both sides:
[tex]\[
-10 = a
\][/tex]
So, [tex]\( a = -10 \)[/tex].
4. Write the Function:
Substitute [tex]\( a = -10 \)[/tex] back into the vertex form equation:
[tex]\[
f(x) = -10(x + 1)^2 + 6
\][/tex]
Therefore, the function [tex]\( f \)[/tex] whose graph is a parabola with vertex [tex]\((-1, 6)\)[/tex] and that passes through the point [tex]\((-2, -4)\)[/tex] is:
[tex]\[
f(x) = -10(x + 1)^2 + 6
\][/tex]
