Write an equation in vertex form of the parabola that has the same shape as the graph of $f(x)=7 x^{2}, but with (5,1) as the vertex.
Answer
\(\boxed{f(x) = 7(x - 5)^2 + 1}\)
Step 1 ：The vertex form of a parabola is given by \(f(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola.
Step 2 ：The 'a' in the vertex form of the equation is the same as the 'a' in the standard form of the equation. In this case, \(a = 7\).
Step 3 ：The vertex of the parabola is given as \((5,1)\), so \(h = 5\) and \(k = 1\).
Step 4 ：Substituting these values into the vertex form gives the equation of the parabola.
Step 5 ：\(a = 7\)
Step 6 ：\(h = 5\)
Step 7 ：\(k = 1\)
Step 8 ：The equation of the parabola in vertex form is \(f(x) = 7(x - 5)^2 + 1\)
Step 9 ：\(\boxed{f(x) = 7(x - 5)^2 + 1}\)
