Unit 9, Lesson 20
Changes over Equal Intervals
Integrated Math 1
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Here is a graph of , where .
Graph of an increasing linear function, x y plane, origin O. Horizontal axis, scale 0 to 10, by 5’s. Vertical axis, scale 0 to 15, by 20’s. The function starts at (0 comma 3) and goes through (1 comma 7) , ( 2 comma 9), (3 comma 11), (4 comma 13) and (5 comma 15).
Here is an expression we can use to find the difference in the values of when the input changes from to .
Does this expression have the same value as what you found in the previous questions? Show your reasoning.
Linear and exponential functions each behave in a particular way every time their input value increases by the same amount.
Take the linear function defined by . The graph of this function has a slope of 5. That means that each time increases by 1, increases by 5. For example, the points and are both on the graph. When increases by 1 (from 7 to 8), increases by 5 (because ). We can show algebraically that this is always true, regardless of what value takes.
Graph of an increasing linear function, x y plane, origin O. Horizontal axis, scale 0 to 10, by 5’s. Vertical axis, scale 0 to 60, by 20’s. The function starts at (0 comma 3) and goes through (7 comma 38) and (8 comma 43). A horizontal green dashed line over green 1 goes from (7 comma 38) for 1 unt, then up to (8 comma 43) with a green 5 next to it for 5 units.
The value of when increases by 1, or , is . Subtracting and , we have:
This tells us that whenever increases by 1, the difference in the output is always 5. In the lesson, we also saw that when increases by an amount other than 1, the output always increases by the same amount if the function is linear.
Now let's look at an exponential function defined by . If we graph , we see that each time increases by 1, the value doubles. We can show algebraically that this is always true, regardless of what value takes.
Graph of an increasing exponential function, x y plane, origin O. Horizontal axis, scale 0 to 10, by 5’s. Vertical axis, scale 0 to 60, by 20’s. The function starts at the vertical axis and goes through (3 comma 8), (4 comma 16) and (5 comma 32).
The value of when increases by 1, or , is . Dividing by , we have:
This means that whenever increases by 1, the value of always increases by a multiple of 2. In the lesson, we also saw that when increases by an amount other than 1, the output always increases by the same factor if the function is exponential.
A linear function always increases (or decreases) by the same amount over equal intervals. An exponential function increases (or decreases) by equal factors over equal intervals.