This lesson continues to investigate the relationship between the parameters \(a\) and \(b\) in the expression \(a \boldcdot b^x\) and a graph representing the function \(f\) given by \(f(x) = a \boldcdot b^x\). Students start by identifying a function represented by a given graph and using the graph to make sense of a situation. They also examine two abstract graphs, with unlabeled axes, and decide which one represents a given situation. This level of abstraction is appropriate at this stage. It gives students an opportunity to apply what they have learned about the relationship between an exponential context and its graph, and to use graphs to better interpret the contexts. In both cases, they rely on their understanding of the connections between the parameters in an exponential expression and the features of an exponential graph (MP7) to answer questions.
- Identify the initial value and growth factor of an exponential function given a graph showing two points with non-consecutive input values.
- Interpret the intersection of the graphs of two functions that represent a situation.
Let’s investigate what we can learn from graphs that represent exponential functions.
Devices that can run Desmos (recommended) or other graphing technology should be avalable as an optional tool for students to select.
- I can explain the meaning of the intersection of the graphs of two functions in terms of the situations they represent.
- When I know two points on a graph of an exponential function, I can write an equation for the function.
An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, \(f(x)=2 \boldcdot 3^x\) defines an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3.