Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand how mod..."

2020-06-01T14:32:45Z

Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand how mod..."

New page

[[1.5 - Introduction to Inverse Trigonometric Functions | Prior Lesson]] | [[2.2 - Solutions to Trigonometric Equations | Next Lesson]]

==Objectives:==

*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==

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:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/qwmcmrfm8i?embed
|width=500
|height=500
|border=0
}}

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.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/nzdnnkruke?embed
|width=500
|height=500
|border=0
}}

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==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

2.1 - Generalized Sinusoidal Functions - Revision history

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand how mod..."