Chapter 10.5, Problem 2E

Solution

To state: Two methods that can be used to model a sinusoid.

The two methods are: write a sine or cosine function and use a graphing calculator

Given information:

The given statement says to describe two methods that can be used to model a sinusoid.

Explanation:

Method 1: To model a sinusoid, write a sine or cosine function to find the values of $a,b,h,\text{ and }k$ for $y=a\cos b\left(x-h\right)+k\text{ or }y=a\sin b\left(x-h\right)+k$ .

Here $\left|a\right|$ is the amplitude, $\frac{2\pi }{\left|b\right|}$ is the period, h is horizontal shift and k is the vertical shift.

First find the maximum value M and the minimum value m . Then identify the vertical shift k .

The value of k is the mean of the maximum and minimum values.

The vertical shift is:

  $\text{Vertical shift}\left(k\right)=\frac{M+m}{2}$

Now decide whether the graph should be modeled by a sine or a cosine function.

When $t=0$ , the graph is at its minimum or maximum or the y -axis at a point, the graph is cosine curve with no horizontal shift.

When the graph crosses the midline on y -axis the graph is sine curve with no horizontal shift. Thus the value of h will be 0.

Now find the amplitude and period.

  $\begin{array}{c}\text{Amplitude}\left(a\right)=\frac{M-m}{2}\\ \text{Period}=\frac{2\pi }{\left|b\right|}\end{array}$

If the graph is a reflection, the value of $a$ will be negative otherwise positive.

Now substitute all the values in the following function:

  $y=a\cos b\left(x-h\right)+k\text{ or }y=a\sin b\left(x-h\right)+k$

Method 2: Another method is to use a graphing calculator that has a sinusoidal regression feature.

The main benefit of this method is that it uses all the data points to find the model.

First of all enter the data in a graphing calculator.

Make a scatter plot.

Perform a sinusoidal regression, because the scatter plot appears sinusoidal.

Now graph the model and the data in the same viewing window.