2.1 - Generalized Sinusoidal Functions
Objectives:[edit]
- Understand how modifying the equation of a trigonometric function will change the period, amplitude, and midline of its graph.
- Write an equation for a trigonometric function given its graph.
Lesson Guide[edit]
Today's lesson looks at transformations of the sine and cosine functions. Students (in 103) should remember this material from Chapter 6. However, this section will serve as a reminder, specifically in the context of trigonometric functions.
The generalized sine and cosine functions have the following forms:
$A\sin(B(x - h)) + k$ and $A\cos(B(x - h)) + k$.
The most basic sine and cosine functions are when $A = 1$, $k =0$, $B =1$, and $h =0$.
Use Desmos to remind students how changing the values of $k$, $A$, and $B$ changes the graph of the function $A\sin(B(x-h))+k$ (set $h=0$ for now). You might have students complete a table, filling in different $k$-, $A$-, and $B$-values and finding the midline, period, and amplitude of the resulting graph. If desired, you can open the graph below and use the value sliders to give them a general idea of the effect of changing each parameter:
Help students to arrive at the following conclusions (you might relate the values back to what they meant in Chapter 6 (for instance, a vertical shift changes the midline of sine and cosine):
In the function $A\sin(B(x - h)) + k$,
- The midline is $y=k$
- The amplitude is $|A|$
- The period is $2\pi/B$
Have students complete Problems 1 and 2 on the worksheet.
Tell students that we would also like to work backwards: given a graph of sine or cosine, we would like to be able to find a trigonometric function that gives the graph. Consider presenting an example like the one below (from Section 9.1) to motivate this process.
A Ferris wheel is $30$ meters in diameter, and is boarded at ground level. The wheel completes one full revolution every $4$ minutes. At time $t=0$, an individual is at the 3:00 position and is ascending. Sketch a graph of $H=f(t)$, where $H$ is the height above ground (in meters) after $t$ minutes.
To find a trigonometric function that gives the graph above, we need to do the following:
- choose a base function (sine or cosine, in this case), determine the period, amplitude, and midline. \\
- Ask students which base function we should choose, and discuss whether to use a sine or cosine function.
- Next, have students find the period, midline, and amplitude of the Ferris wheel graph, and use these to calculate $A$, $B$, and $k$.
Remind students that after they have what they think is the correct equation, they should double-check by testing some values and making sure they actually fall on the curve.
You can also use this example to introduce horizontal shifts.
Have students complete Problem 3 in the workbook.
You can now either introduce students to horizontal shifts or let them explore the concept on their own by having them work through Problems 4, 5, and 6.
Comments[edit]
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