To determine the equation described in the table, we can observe the relationship between the values of \(x\) and \(y\). The table shows that as \(x\) increases, \(y\) increases by 1 for each increase of 1 in \(x\). This indicates a linear relationship.
We can notice that when \(x = 0\), \(y = -10\). This gives us a starting point. The general form of a linear equation is:
\[ f(x) = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept.
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Finding the slope (m): The slope \(m\) can be found from the change in \(y\) over the change in \(x\): \[ m = \frac{\Delta y}{\Delta x} = \frac{-9 - (-10)}{1 - 0} = \frac{1}{1} = 1 \]
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Finding the y-intercept (b): From the starting point when \(x = 0\), we see that \(y = -10\), so: \[ b = -10 \]
Putting it all together, we find:
\[ f(x) = 1x - 10 \]
This can also be simply written as:
\[ f(x) = x - 10 \]
So, in function notation, the answer is:
f(x) = 1x + (-10)
or more succinctly:
f(x) = x - 10
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