Question

enter the correct answer in the box.
write the function in vertex form for this parabola.
(image of a parabola with focus at (3, -2.5) and a grid)
f(x) =

Explanation:

Step1: Identify the vertex

The vertex of the parabola is the point where the parabola changes direction. From the graph, the vertex (h, k) is at (3, -3) (since the focus is at (3, -2.5), and the vertex is directly below the focus for a parabola that opens upward, and the distance between the vertex and focus is \( p \)).

Step2: Determine the value of \( p \)

The focus is at (3, -2.5) and the vertex is at (3, -3). The distance \( p \) is the distance between the vertex and the focus. So \( p = -2.5 - (-3) = 0.5=\frac{1}{2} \).

Step3: Recall the vertex form of a parabola

The vertex form of a parabola that opens upward (since the parabola opens upward as it has a minimum point) is \( f(x)=a(x - h)^2 + k \), and for a parabola, \( a=\frac{1}{4p} \).

Step4: Calculate \( a \)

Since \( p=\frac{1}{2} \), then \( a = \frac{1}{4\times\frac{1}{2}}=\frac{1}{2} \).

Step5: Substitute \( h \), \( k \), and \( a \) into the vertex form

We have \( h = 3 \), \( k=-3 \), and \( a=\frac{1}{2} \). So the vertex form is \( f(x)=\frac{1}{2}(x - 3)^2-3 \).

Answer:

\( \frac{1}{2}(x - 3)^2 - 3 \)