To determine the equation that describes the function shown in the mapping diagram, we need to analyze the relationship between the values of \(x\) and \(y\).
We can see the points paired:
- \(x = 0 \rightarrow y = -8\)
- \(x = 1 \rightarrow y = -7.5\)
- \(x = 2 \rightarrow y = -7\)
- \(x = 3 \rightarrow y = -6.5\)
Now, let's analyze the changes in values. The output \(y\) increases by \(0.5\) for each increase in \(x\) by \(1\). This indicates a linear relationship.
The change in \(y\) per unit change in \(x\) can be captured by the slope \( m = 0.5 \). The equation for a linear function typically follows the form:
\[ y = mx + b \]
Where:
- \(m\) is the slope.
- \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).
Plugging in what we have:
- The slope \(m = 0.5\),
- When \(x = 0\), \(y = -8\), thus \(b = -8\).
The equation can be expressed as:
\[ y = 0.5x - 8 \]
Now, if we want to express \(y\) in terms of \(x\), we get:
\[ f(x) = 0.5x - 8 \]
However, looking at the responses provided, the closest match is:
- \( f(x) = \frac{1}{2}x - 8 \)
Thus, the answer is:
f(x) = \frac{1}{2} x - 8.
1 answer