Question

Use input-output tables to determine which graph accurately represents the function f(x) = 1/2x + 3 with inputs of -4, -2, 0, 2, and 4.

Answer 1

To determine which graph accurately represents the function \( f(x) = \frac{1}{2}x + 3 \) with the specified inputs, we will first calculate the output values for each input using the function.

Let's create an input-output table for \( f(x) \):

1. For \( x = -4 \):
\[
f(-4) = \frac{1}{2}(-4) + 3 = -2 + 3 = 1
\]

2. For \( x = -2 \):
\[
f(-2) = \frac{1}{2}(-2) + 3 = -1 + 3 = 2
\]

3. For \( x = 0 \):
\[
f(0) = \frac{1}{2}(0) + 3 = 0 + 3 = 3
\]

4. For \( x = 2 \):
\[
f(2) = \frac{1}{2}(2) + 3 = 1 + 3 = 4
\]

5. For \( x = 4 \):
\[
f(4) = \frac{1}{2}(4) + 3 = 2 + 3 = 5
\]

Now that we have calculated the output values, let's summarize the input-output pairs:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & 1 \\
-2 & 2 \\
0 & 3 \\
2 & 4 \\
4 & 5 \\
\hline
\end{array}
\]

The pairs are:
- (-4, 1)
- (-2, 2)
- (0, 3)
- (2, 4)
- (4, 5)

To find the correct graph, plot these points on the graph, and ensure that it best fits the linear equation \( f(x) = \frac{1}{2}x + 3 \). The line should rise gradually due to the slope of \( \frac{1}{2} \), indicating that for every 2 units in the x-direction, the y-value increases by 1 unit.

You can now compare these points with the provided graph options to identify which one passes through all or most of these points.