Question

The table below represents a linear situation.

x f(x)
-1 3
0 7
1 11
2 15

Use this table to construct the function that it represents.
A.
f(x) = x + 7
B.
f(x) = 7x - 4
C.
f(x) = -4x - 1
D.
f(x) = 4x + 7

Answer 1

To determine the function that fits the data in the table, we first need to identify the pattern in the given values. The table provides the following points:

\[
\begin{align*}
(-1, 3) \\
(0, 7) \\
(1, 11) \\
(2, 15) \\
\end{align*}
\]

We need to find the slope (m) of the linear function using two points from the table. We can use the points (0, 7) and (1, 11) to compute the slope.

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (0, 7) and (1, 11):

\[
m = \frac{11 - 7}{1 - 0} = \frac{4}{1} = 4
\]

Now that we have the slope, we can use the point-slope form of a linear equation which is:

\[
f(x) = mx + b
\]

We know \( m = 4 \). Next, we will find \( b \) by using one of the points from the table, for example (0, 7):

\[
7 = 4(0) + b \implies b = 7
\]

Now we can construct the linear function:

\[
f(x) = 4x + 7
\]

Now let's check which of the provided options matches our function:

- A: \( f(x) = x + 7 \)
- B: \( f(x) = 7x - 4 \)
- C: \( f(x) = -4x - 1 \)
- D: \( f(x) = 4x + 7 \)

The correct answer is:

**D. \( f(x) = 4x + 7 \)**