The goal of this lesson is to introduce the midline and amplitude of trigonometric functions in context. The midline is given by the average of the maximum and minimum values taken by the function, while the amplitude is the length between the maximum value and the midline or, equivalently, the length between the midline and the minimum value. The function \(f\) given by \(f(x) = \cos(x)\) has a midline of \(y=0\) (since the maximum value is 1 and the minimum value is -1) and an amplitude of 1. The function \(g(x) = 5\sin(x) -1\) has a midline of \(y=\text-1\) and an amplitude of 5. In general, the function \(h\) given by \(h(x) = a\cos(x) + b\) has a midline of \(y=b\) and an amplitude of \(|a|\).
The midline is a new feature of trigonometric functions. The amplitude, on the other hand, relates to work students have done in a previous unit using vertical scale factors. For example, the amplitude of the function \(g\) given by \(g(x) = 5 \sin(x)\) is 5. The graph of \(g\) is the graph of \(h(x) = \sin(x)\) after it has been stretched vertically by a factor of 5. Said another way, the outputs of \(g(x)\) are 5 times farther from the \(x\)-axis than the outputs of \(h(x)\) for the same input values.
Students reason abstractly and quantitatively when they interpret a trigonometric function in the context of a rotating windmill blade (MP2). They represent the function in three different ways including a table, a graph, and an equation. Students make use of repeated reasoning to determine the effect of different parameters on the amplitude and midline of trigonometric functions (MP8).
Throughout this lesson, students should have access to their unit circle and graph of sine and cosine displays.
- Create a trigonometric function to represent a situation where either the amplitude or midline is not 1.
- Describe (in writing) how changing the amplitude or midline transforms the graph of a trigonometric function.
- Let's transform the graphs of trigonometric functions
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can write a trigonometric function to represent situations with different amplitudes and midlines.
The maximum distance of the values of a periodic function above or below the midline.
The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose \(y\)-coordinate is that value.