To find the equation of the quadratic function in vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], we follow these steps:
1. Identify the vertex ([tex]\(h\)[/tex], [tex]\(k\)[/tex]):
The vertex of the function is at the point where the function reaches its maximum or minimum value. From the given table, we see that the function reaches its maximum value at [tex]\(x = 8\)[/tex] with [tex]\(f(x) = 6\)[/tex]. Therefore, the vertex is [tex]\( (h, k) = (8, 6) \)[/tex].
2. Substitute the vertex into the vertex form equation:
The equation now looks like [tex]\( f(x) = a(x-8)^2 + 6 \)[/tex].
3. Determine the value of [tex]\(a\)[/tex]:
To find the value of [tex]\(a\)[/tex], use another point from the table. Let's use the point [tex]\((6, -2)\)[/tex].
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( f(x) = -2 \)[/tex] into the equation:
[tex]\[
-2 = a(6 - 8)^2 + 6
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
[tex]\[
-2 = a(-2)^2 + 6
\][/tex]
[tex]\[
-2 = 4a + 6
\][/tex]
Subtract 6 from both sides:
[tex]\[
-2 - 6 = 4a
\][/tex]
[tex]\[
-8 = 4a
\][/tex]
Divide both sides by 4:
[tex]\[
a = -2
\][/tex]
5. Write the final equation:
Substitute [tex]\(a = -2\)[/tex], [tex]\(h = 8\)[/tex], and [tex]\(k = 6\)[/tex] into the vertex form equation:
[tex]\[
f(x) = -2(x-8)^2 + 6
\][/tex]
So, the equation of the function in vertex form is:
[tex]\[
f(x) = -2(x-8)^2 + 6
\][/tex]
