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Sagot :
To understand the effect of doubling the interval length on the average rate of change of a linear function, let's analyze the concept of the average rate of change and how it applies to linear functions.
### Step-by-Step Solution:
1. Definition of Average Rate of Change:
The average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \][/tex]
2. Linear Function Properties:
For a linear function of the form [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept, the slope [tex]\( m \)[/tex] represents the rate of change. This slope is constant for any two distinct points on the linear function.
3. Impact of Changing the Interval:
Let's consider doubling the interval length. If the original interval is [tex]\([a, b]\)[/tex] with length [tex]\( b - a \)[/tex], doubling this interval gives us a new interval [tex]\([a, a + 2(b - a)]\)[/tex].
[tex]\[ \text{Original Interval Length} = b - a \][/tex]
[tex]\[ \text{New Interval Length} = 2(b - a) = b - a + (b - a) \][/tex]
4. Computing the Average Rate of Change for Original and New Intervals:
For the original interval [tex]\([a, b]\)[/tex], the average rate of change is:
[tex]\[ \text{Original Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \][/tex]
Since [tex]\( f(x) = mx + c \)[/tex], we get [tex]\( f(b) = mb + c \)[/tex] and [tex]\( f(a) = ma + c \)[/tex], so:
[tex]\[ \text{Original Average Rate of Change} = \frac{mb + c - (ma + c)}{b - a} = \frac{mb - ma}{b - a} = \frac{m(b - a)}{b - a} = m \][/tex]
For the new interval [tex]\([a, a + 2(b - a)]\)[/tex], the average rate of change is:
[tex]\[ \text{New Average Rate of Change} = \frac{f(a + 2(b - a)) - f(a)}{(a + 2(b - a)) - a} \][/tex]
Again, using [tex]\( f(x) = mx + c \)[/tex], we have [tex]\( f(a + 2(b - a)) = m(a + 2(b - a)) + c \)[/tex]:
[tex]\[ \text{New Average Rate of Change} = \frac{m(a + 2(b - a)) + c - (ma + c)}{2(b - a)} \][/tex]
Simplifying the numerator, we get:
[tex]\[ \text{New Average Rate of Change} = \frac{ma + 2m(b - a) + c - ma - c}{2(b - a)} = \frac{2m(b - a)}{2(b - a)} = m \][/tex]
5. Conclusion:
Since the average rate of change [tex]\( m \)[/tex] remains the same regardless of the interval length, we can conclude that doubling the interval length has no effect on the average rate of change of a linear function.
Thus, the correct choice is:
- It has no effect on the average rate of change.
So, the answer is:
It has no effect on the average rate of change.
### Step-by-Step Solution:
1. Definition of Average Rate of Change:
The average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \][/tex]
2. Linear Function Properties:
For a linear function of the form [tex]\( f(x) = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept, the slope [tex]\( m \)[/tex] represents the rate of change. This slope is constant for any two distinct points on the linear function.
3. Impact of Changing the Interval:
Let's consider doubling the interval length. If the original interval is [tex]\([a, b]\)[/tex] with length [tex]\( b - a \)[/tex], doubling this interval gives us a new interval [tex]\([a, a + 2(b - a)]\)[/tex].
[tex]\[ \text{Original Interval Length} = b - a \][/tex]
[tex]\[ \text{New Interval Length} = 2(b - a) = b - a + (b - a) \][/tex]
4. Computing the Average Rate of Change for Original and New Intervals:
For the original interval [tex]\([a, b]\)[/tex], the average rate of change is:
[tex]\[ \text{Original Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \][/tex]
Since [tex]\( f(x) = mx + c \)[/tex], we get [tex]\( f(b) = mb + c \)[/tex] and [tex]\( f(a) = ma + c \)[/tex], so:
[tex]\[ \text{Original Average Rate of Change} = \frac{mb + c - (ma + c)}{b - a} = \frac{mb - ma}{b - a} = \frac{m(b - a)}{b - a} = m \][/tex]
For the new interval [tex]\([a, a + 2(b - a)]\)[/tex], the average rate of change is:
[tex]\[ \text{New Average Rate of Change} = \frac{f(a + 2(b - a)) - f(a)}{(a + 2(b - a)) - a} \][/tex]
Again, using [tex]\( f(x) = mx + c \)[/tex], we have [tex]\( f(a + 2(b - a)) = m(a + 2(b - a)) + c \)[/tex]:
[tex]\[ \text{New Average Rate of Change} = \frac{m(a + 2(b - a)) + c - (ma + c)}{2(b - a)} \][/tex]
Simplifying the numerator, we get:
[tex]\[ \text{New Average Rate of Change} = \frac{ma + 2m(b - a) + c - ma - c}{2(b - a)} = \frac{2m(b - a)}{2(b - a)} = m \][/tex]
5. Conclusion:
Since the average rate of change [tex]\( m \)[/tex] remains the same regardless of the interval length, we can conclude that doubling the interval length has no effect on the average rate of change of a linear function.
Thus, the correct choice is:
- It has no effect on the average rate of change.
So, the answer is:
It has no effect on the average rate of change.
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