Question

determine the quadratic function of the form $f(x)=a(x - h)^2 + k$ whose graph is given on the right.
$f(x)=\square$
(do not simplify.)

Explanation:

Step1: Identify vertex \((h,k)\)

From the graph, the vertex of the parabola is at \((2, 6)\), so \(h = 2\) and \(k = 6\). The equation becomes \(f(x)=a(x - 2)^2+6\).

Step2: Use a point to find \(a\)

The graph passes through the point \((0, 1)\) (from the y - intercept or the given point on the graph). Substitute \(x = 0\) and \(f(x)=1\) into the equation:
\[1=a(0 - 2)^2+6\]
\[1 = 4a+6\]

Step3: Solve for \(a\)

Subtract 6 from both sides:
\[1-6=4a\]
\[- 5 = 4a\]
Divide both sides by 4:
\[a=-\frac{5}{4}\]

Step4: Write the function

Substitute \(a =-\frac{5}{4}\), \(h = 2\) and \(k = 6\) into \(f(x)=a(x - h)^2+k\):
\[f(x)=-\frac{5}{4}(x - 2)^2+6\]

Answer:

\(f(x)=-\frac{5}{4}(x - 2)^2+6\)