To determine the equation of the linear function that models the relationship shown in the table, we first need to identify the values for \(x\) and \(y\) to see how they relate.
From the table you provided:
- For \(y = 4\), \(x = 38\)
- For \(y = 6\), \(x = 58\)
Now, we can find the slope (\(m\)) of the linear function using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((38, 4)\) and \((58, 6)\):
\[ m = \frac{6 - 4}{58 - 38} = \frac{2}{20} = \frac{1}{10} \]
Now we can use the point-slope form of a linear equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
We can use the point \( (38, 4) \) for \( x_1 \) and \( y_1 \):
\[ y - 4 = \frac{1}{10}(x - 38) \]
Now we will rearrange to get it in slope-intercept form \( y = mx + b \):
\[ y - 4 = \frac{1}{10}x - \frac{38}{10} \] \[ y - 4 = \frac{1}{10}x - 3.8 \] \[ y = \frac{1}{10}x - 3.8 + 4 \] \[ y = \frac{1}{10}x + 0.2 \]
Now, looking at the choices provided:
- \(y = -10x + 78\)
- \(y = 10x - 2\)
- \(y = x + 34\)
- \(y = 10x + 2\)
None of these choices seem to match with the derived equation \(y = \frac{1}{10}x + 0.2\).
Let's analyze the original data points again to see if any calculation mistakes were made. The data points suggest:
- if you assume the first \(x\) corresponds to the first \(y\),
- and the second corresponds in the same sequence, it wouldn't match the expected formats.
I'll double-check those explicit relationships for \(y\) as \(4, 6\) corresponding to \(x\) as \(38, 58\), which appears correct.
However, if we were to assume a systematic option is at play, you can cross-check the expected outputs from standard options with given numbers,
It seems we need to double verify \(y\) equations were expected per format.
Check if a possibility of linear transformations revealed errors in linear representation.
Given that the \(x\) conversions align up with linear representations, accurate pondering of all segments linearly show there's no fixed immediate output.
If any candidates missed clarity or dynamic transformations forgot fundamental slope interactions, feel free to share values needing portrayal.