To determine which equation describes the function shown in the mapping diagram, we need to find the relationship between the \(x\) values and the corresponding \(y\) values.
The \(x\) values are:
- \(0\)
- \(1\)
- \(2\)
- \(3\)
The corresponding \(y\) values are:
- \(-8\)
- \(-7.5\)
- \(-7\)
- \(-6.5\)
Let's analyze the changes in \(y\) as \(x\) increases:
- When \(x = 0\), \(y = -8\)
- When \(x = 1\), \(y = -7.5\) (increased by \(0.5\))
- When \(x = 2\), \(y = -7\) (increased by \(0.5\))
- When \(x = 3\), \(y = -6.5\) (increased by \(0.5\))
This shows that for every increase of \(1\) in \(x\), \(y\) increases by \(0.5\). Thus, we can establish a linear relationship:
The slope \(m\) can be calculated as: \[ m = \frac{\Delta y}{\Delta x} = \frac{0.5}{1} = 0.5 \]
Now, using the slope-intercept form \(y = mx + b\), we can determine \(b\) (the y-intercept). We know that when \(x = 0\), \(y = -8\): \[ -8 = 0.5(0) + b \] Thus, \[ b = -8 \]
Now we can write the equation: \[ y = 0.5x - 8 \]
This can be rewritten in terms of \(f(x)\) as: \[ f(x) = 0.5x - 8 \]
Now, looking at the choices provided, the correct one is: \[ f(x) = \frac{1}{2}x - 8 \]
So the answer is: f(x) = 1/2 x - 8.
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