Changes over Equal Intervals
In this lesson, students work to show that linear functions change by equal differences over equal intervals while exponential functions grow by equal factors over equal intervals. When the interval length is 1, the equal difference is the slope of a linear function. When the interval length is more than 1, the difference is the length of the interval times the slope. For an exponential function \(f\) given by \(f(x) = a \boldcdot b^x\), when the interval length is 1, the factor of growth is \(b\). If the interval length is a whole number \(n\), then the factor of growth is \(b^n\).
In order to justify why linear functions change by equal differences over equal intervals and exponential functions change by equal factors over equal intervals, students observe this structure repeatedly for specific numbers and then generalize this reasoning to apply to all intervals (MP8). Along the way, they need to recognize and use properties of exponents, which is an example of noticing and making use of structure (MP7).
- Calculate rates of change of functions given graphs, equations, or tables.
- Use rates of change and show that, for any equal intervals of the independent variable, an exponential function always increases or decreases by an equal factor.
- Use rates of change to show that, for any equal intervals of the independent variable, a linear function always changes by an equal difference.
Let's explore how linear and exponential functions change over equal intervals.
- I can calculate rates of change of functions given graphs, equations, or tables.
- I can use rates of change to describe how a linear function and an exponential function change over equal intervals.
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