Question

write an equation for the parabola that has the given vertex and passes through the given point.
vertex
(5,1)
point
(3,13)
f(x) = ?(x + )² +

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is $f(x)=a(x - h)^2 + k$, where \((h,k)\) is the vertex. Given vertex \((5,1)\), so \(h = 5\), \(k = 1\). Then the equation becomes \(f(x)=a(x - 5)^2+1\), which can be rewritten as \(f(x)=a(x+(- 5))^2 + 1\).

Step2: Substitute the point \((3,13)\) into the equation

Substitute \(x = 3\) and \(f(x)=13\) into \(f(x)=a(x - 5)^2+1\):
\[13=a(3 - 5)^2+1\]

Step3: Solve for \(a\)

Simplify the equation:
\[13=a(-2)^2+1\]
\[13 = 4a+1\]
Subtract 1 from both sides:
\[12 = 4a\]
Divide both sides by 4:
\[a = 3\]

Answer:

\(f(x)=3(x + (-5))^2 + 1\) (or simplified as \(f(x)=3(x - 5)^2 + 1\)) with the blanks filled as \(3\), \(-5\), \(1\) respectively.