Question

which choice is the correct vertex - form of the quadratic function graphed below. a) ( f(x)=(x - 4)^{2}+1 ) b) ( f(x)=\frac{1}{2}(x - 4)^{2}+1 ) c) ( f(x)=2(x + 4)^{2}+1 ) d) ( f(x)=2(x - 4)^{2}+1 )

Explanation:

Step1: Recall vertex - form of a quadratic function

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where the vertex of the parabola is the point $(h,k)$.

Step2: Analyze the vertex of the given parabola

From the graph, the vertex of the parabola is at the point $(4, - 3)$.

Step3: Check each option

For option A: $f(x)=(x - 4)^2+1$, the vertex is $(4,1)$.
For option B: $f(x)=\frac{1}{2}(x - 4)^2+1$, the vertex is $(4,1)$.
For option C: $f(x)=2(x + 4)^2+1$, the vertex is $(-4,1)$.
For option D: $f(x)=2(x - 4)^2-3$, the vertex is $(4,-3)$.

Answer:

D. $f(x)=2(x - 4)^2-3$