To determine the correct equation of the quadratic function, we need to use the given information:
1. The vertex of the quadratic function is [tex]\((5, 3)\)[/tex].
2. The function passes through the point [tex]\((-1, -9)\)[/tex].
The vertex form of a quadratic function is written as:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Given the vertex [tex]\((5, 3)\)[/tex], we have:
[tex]\[ f(x) = a(x - 5)^2 + 3 \][/tex]
To find the value of [tex]\(a\)[/tex], we substitute the coordinates of the given point [tex]\((-1, -9)\)[/tex] into the equation and solve for [tex]\(a\)[/tex]:
[tex]\[ -9 = a(-1 - 5)^2 + 3 \][/tex]
First, simplify the expression inside the parentheses:
[tex]\[ -1 - 5 = -6 \][/tex]
Next, substitute [tex]\(-6\)[/tex] back into the equation:
[tex]\[ -9 = a(-6)^2 + 3 \][/tex]
Simplify [tex]\((-6)^2\)[/tex]:
[tex]\[ (-6)^2 = 36 \][/tex]
Now the equation becomes:
[tex]\[ -9 = 36a + 3 \][/tex]
Subtract 3 from both sides to isolate the term with [tex]\(a\)[/tex]:
[tex]\[ -12 = 36a \][/tex]
Divide both sides by 36 to solve for [tex]\(a\)[/tex]:
[tex]\[ a = -\frac{1}{3} \][/tex]
Now, substitute [tex]\(a = -\frac{1}{3}\)[/tex] back into the vertex form of the equation:
[tex]\[ f(x) = -\frac{1}{3}(x - 5)^2 + 3 \][/tex]
Thus, the equation of the function is:
[tex]\[ f(x) = -\frac{1}{3}(x - 5)^2 + 3 \][/tex]
Comparing this with the given options, we find that the correct answer is:
[tex]\[ \boxed{f(x) = -\frac{1}{3}(x - 5)^2 + 3} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{4} \][/tex]
