Exercise 1 Page 488

We will find the period and amplitude of the given sine function. Then, we will graph the function and describe the graph of g as a transformation of its parent function. Let's do these things one at a time.

Finding Period and Amplitude

Let's consider the general form of a sine function. y= a sin bx Here | a| is the amplitude, | b| is the number of cycles in the interval from 0 to 2π, and 2π| b| is the period of the function. Let's now consider the given function. g(x)=1/4sin x ⇔ g(x)= 1/4 sin 1xIn this equation we have that a= 14 and b= 1. Since | 14|= 14, the amplitude of the graph of the function is 14. This means that the maximum is 14 and the minimum is - 14. Amplitude:& 1/4 Maximum:& 1/4 Minimum:& - 1/4 With b= 1, we know that there is 1 cycle from 0 to 2π. This means that the period of function g is no different from the period of the parent function, the sine function. Therefore, the period of this function is 2π.

Sketching the Graph

To graph the given function, we should first identify the points where the x- intercepts, maximum values, and minimum values occur. These points can be identified for a sine function as follows. x- intercepts:& ( 0,0), ( 1/2*2π/b,0) , ( 2π/b,0) [1.1em] Maximum:& ( 1/4*2π/b,a) [1.1em] Minimum:& ( 3/4*2π/b, - a ) By substituting a= 14 and b=1, we can evaluate these points. x- intercepts:& ( 0,0), ( π,0), ( 2π,0) [1.1em] Maximum:& ( π/2,0.25) [1.1em] Minimum:& ( 3π/2, - 0.25 ) We can graph one cycle of the function by substituting calculated points and connecting them with a smooth curve. Let's do it!

Finally, we can obtain the desired graph by extending the pattern along the x-axis.

Describing the Graph as a Transformation of f(x)=sin(x)

To describe the transformation being applied to the parent function, let's start by recalling two possible transformations of the function f(x)=sin(x).

Now consider our original equation. g(x)= 1/4 sin ( 1x ) We can see that a= 14, and b= 1. Therefore, the graph of g is a vertical shrink by a factor of 14 of the graph of f(x)=sin(x).