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Answer :
To determine if a function represented by a set of data points is linear, quadratic, or exponential, we can examine the differences between consecutive [tex]\( y \)[/tex]-values.
Here are the data points given:
- [tex]\( (3, -20.7) \)[/tex]
- [tex]\( (4, -24.5) \)[/tex]
- [tex]\( (5, -28.3) \)[/tex]
- [tex]\( (6, -32.1) \)[/tex]
- [tex]\( (7, -35.9) \)[/tex]
Step-by-Step Analysis:
1. Calculate the Differences Between Consecutive [tex]\( y \)[/tex]-Values:
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 4 \)[/tex] and [tex]\( x = 3 \)[/tex]: [tex]\(-24.5 - (-20.7) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 5 \)[/tex] and [tex]\( x = 4 \)[/tex]: [tex]\(-28.3 - (-24.5) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 6 \)[/tex] and [tex]\( x = 5 \)[/tex]: [tex]\(-32.1 - (-28.3) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 7 \)[/tex] and [tex]\( x = 6 \)[/tex]: [tex]\(-35.9 - (-32.1) = -3.8\)[/tex]
2. Check if the Differences Are Constant:
When we observe these differences, they are all equal to approximately [tex]\(-3.8\)[/tex]. This constant rate of change indicates that the function is linear.
Conclusion:
Since the differences between consecutive [tex]\( y \)[/tex]-values are constant, the function is linear.
Here are the data points given:
- [tex]\( (3, -20.7) \)[/tex]
- [tex]\( (4, -24.5) \)[/tex]
- [tex]\( (5, -28.3) \)[/tex]
- [tex]\( (6, -32.1) \)[/tex]
- [tex]\( (7, -35.9) \)[/tex]
Step-by-Step Analysis:
1. Calculate the Differences Between Consecutive [tex]\( y \)[/tex]-Values:
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 4 \)[/tex] and [tex]\( x = 3 \)[/tex]: [tex]\(-24.5 - (-20.7) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 5 \)[/tex] and [tex]\( x = 4 \)[/tex]: [tex]\(-28.3 - (-24.5) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 6 \)[/tex] and [tex]\( x = 5 \)[/tex]: [tex]\(-32.1 - (-28.3) = -3.8\)[/tex]
- Difference between [tex]\( y \)[/tex] at [tex]\( x = 7 \)[/tex] and [tex]\( x = 6 \)[/tex]: [tex]\(-35.9 - (-32.1) = -3.8\)[/tex]
2. Check if the Differences Are Constant:
When we observe these differences, they are all equal to approximately [tex]\(-3.8\)[/tex]. This constant rate of change indicates that the function is linear.
Conclusion:
Since the differences between consecutive [tex]\( y \)[/tex]-values are constant, the function is linear.
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