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 \( f(x) \):
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For \( x = -4 \): \[ f(-4) = \frac{1}{2}(-4) + 3 = -2 + 3 = 1 \]
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For \( x = -2 \): \[ f(-2) = \frac{1}{2}(-2) + 3 = -1 + 3 = 2 \]
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For \( x = 0 \): \[ f(0) = \frac{1}{2}(0) + 3 = 0 + 3 = 3 \]
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For \( x = 2 \): \[ f(2) = \frac{1}{2}(2) + 3 = 1 + 3 = 4 \]
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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.