First, let's convert the y-values into improper fractions for easier calculations:
- For \( y = 7.5 \), it can be written as \( \frac{15}{2} \).
- For \( y = -7 \), it remains as \( -7 \).
- For \( y = -6.5 \), it can be written as \( \frac{-13}{2} \).
Now, we have the pairs in the mapping:
- \( (0, -8) \)
- \( (1, \frac{15}{2}) \)
- \( (2, -7) \)
- \( (3, \frac{-13}{2}) \)
Now, let's check which of the given options might fit these points.
Option 1: \( F(x) = \frac{1}{2}x - 8 \)
- For \( x = 0 \): \( F(0) = \frac{1}{2}(0) - 8 = -8 \) ✓
- For \( x = 1 \): \( F(1) = \frac{1}{2}(1) - 8 = \frac{1}{2} - 8 = \frac{1}{2} - \frac{16}{2} = \frac{-15}{2} \) ✗
- For \( x = 2 \): \( F(2) = \frac{1}{2}(2) - 8 = 1 - 8 = -7 \) ✓
- For \( x = 3 \): \( F(3) = \frac{1}{2}(3) - 8 = \frac{3}{2} - 8 = \frac{3}{2} - \frac{16}{2} = \frac{-13}{2} \) ✓
Option 2: \( F(x) = 8x - \frac{1}{2} \)
- For \( x = 0 \): \( F(0) = 8(0) - \frac{1}{2} = -\frac{1}{2} \) ✗
- For \( x = 1 \): \( F(1) = 8(1) - \frac{1}{2} = 8 - \frac{1}{2} = \frac{16}{2} - \frac{1}{2} = \frac{15}{2} \) ✓
- For \( x = 2 \): \( F(2) = 8(2) - \frac{1}{2} = 16 - \frac{1}{2} = \frac{32}{2} - \frac{1}{2} = \frac{31}{2} \) ✗
- For \( x = 3 \): \( F(3) = 8(3) - \frac{1}{2} = 24 - \frac{1}{2} = \frac{48}{2} - \frac{1}{2} = \frac{47}{2} \) ✗
Option 3: \( F(y) = 8x - \frac{1}{2} \)
This option describes \( F(y) \) rather than \( F(x) \), so we will not evaluate this.
Option 4: \( F(y) = \frac{1}{2}x - 8 \)
Again, this describes \( F(y) \), which we will not evaluate.
Conclusion
Among the relevant options, only Option 1 aligns with the point for \( x = 0, 2, \) and \( 3\), and aligns with the overall pattern of growth/decline with parts unsuitable for Option 2 or options describing \( F(y) \).
Therefore, the best representation of the function based on the mapping diagram is:
1. \( F(x) = \frac{1}{2}x - 8 \).
1 answer