Answered: The following graph shows at least one…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter8: Sequences And Series

Section: Chapter Questions

Problem 11T

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Question

The following graph shows at least one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph. If you are using a graphing calculator, graph your equation to verify that it is correct.

Expert Solution

Step 1

The function is continuous so, it can be modelled using sine or cosine function.

Using sine function:

Let the function be $f (x) = A \sin (B x + C) + D$ .

The mid-line occurs at $y = 2$ as maximum is $y = 5$ and minimum is $y = - 1$ , so, we have $D = 2$ .

The height above the mid-line is 3, so, $A = 3$ .

Thus, function becomes $f (x) = 3 \sin (B x + C) + 2$ .

As,

$f (0) = 2$

$\Rightarrow 3 \sin (C) + 2 = 2 \Rightarrow 3 \sin (C) = 0 \Rightarrow \sin (C) = 0 \Rightarrow C = π$

So, function becomes $f (x) = 3 \sin (B x + π) + 2$ .

As,

$f (\frac{1}{2}) = - 1$

$\Rightarrow 3 \sin (\frac{B}{2} + π) + 2 = - 1$

$\Rightarrow 3 \sin (\frac{B}{2} + π) = - 3$

$\Rightarrow \sin (\frac{B}{2} + π) = - 1 \Rightarrow \frac{B}{2} + π = \frac{3 π}{2} \Rightarrow \frac{B}{2} = \frac{π}{2} \Rightarrow B = π$

Thus, the function becomes $f (x) = 3 \sin (πx + π) + 2$ .

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