Question

The table of values for function f(x)
is shown. (14,-55) (-12,-34) (-10,-14) (-8,-10) (-6,-7) (-4,-10) (-2,-19) What is the equation of
f(x) in vertex form?

Answer 1

To find the equation of the function \( f(x) \) in vertex form, we can use the general form of a quadratic function, which is:

\[
f(x) = a(x-h)^2 + k
\]

where \( (h, k) \) is the vertex of the parabola.

**Step 1: Identify the vertex**

To identify the vertex, we will look for the point where the function changes direction; usually, it is a local minimum or maximum. The given points in the table are:

- (14, -55)
- (-12, -34)
- (-10, -14)
- (-8, -10)
- (-6, -7)
- (-4, -10)
- (-2, -19)

From these points, let's analyze the values of \( y \):

- The minimum value in this dataset occurs at \( (-10, -14) \), and since it is the lowest point listed, we can assume it to be the vertex.

Thus, the vertex \( (h, k) \) is \( (-10, -14) \).

**Step 2: Determine the value of \( a \)**

We can use one of the other points to determine the value of \( a \) in the vertex form. Let's use the point \( (-8, -10) \):

\[
f(x) = a(x + 10)^2 - 14
\]

Substituting \( x = -8 \) and \( f(-8) = -10 \):

\[
-10 = a(-8 + 10)^2 - 14
\]
\[
-10 = a(2)^2 - 14
\]
\[
-10 = 4a - 14
\]
\[
4 = 4a
\]
\[
a = 1
\]

**Step 3: Write the final equation**

Thus, the equation of \( f(x) \) in vertex form is:

\[
f(x) = 1(x + 10)^2 - 14
\]

We can simplify this to:

\[
f(x) = (x + 10)^2 - 14
\]

This is the vertex form of the function \( f(x) \).