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 b x

Here

∣ a ∣

is the amplitude,

∣ b ∣

is the number of cycles in the interval from

0

to

2 π,

and

\frac{2 π}{∣ b ∣}

is the period of the function. Let's now consider the given function.

g (x) = \frac{1}{4} sin x \Leftrightarrow g (x) = \frac{1}{4} sin 1 x

In this equation we have that

a = \frac{1}{4}

and

b = 1 .

Since

∣ \frac{1}{4} ∣ = \frac{1}{4},

the amplitude of the graph of the function is

\frac{1}{4} .

This means that the maximum is

\frac{1}{4}

and the minimum is

- \frac{1}{4} .

Amplitude : Maximum : Minimum : - \frac{1}{4} - \frac{1}{4} - \frac{1}{4}

With

b = 1,

we know that there is

1

cycle from

0

to

2 π .

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

2 π .

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 : Maximum : Minimum : (0, 0), (\frac{1}{2} \cdot \frac{2 π}{b}, 0), (\frac{2 π}{b}, 0) (\frac{1}{4} \cdot \frac{2 π}{b}, a) (\frac{3}{4} \cdot \frac{2 π}{b}, - a)

By substituting

a = \frac{1}{4}

and

b = 1,

we can evaluate these points.

x - intercepts : Maximum : Minimum : (0, 0), (π, 0), (2 π, 0) (\frac{π}{2}, 0.25) (\frac{3 π}{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 $f (x) = sin (x) .$

Function	Transformation of the Graph of $f (x) = sin (x)$
$g (x) = a sin (x)$	Vertical Stretch or Shrink by a factor of $a$ .
$g (x) = sin (b x)$	Horizontal Stretch or Shrink by a factor of $\frac{1}{b}$ .

Now consider our original equation.

g (x) = \frac{1}{4} sin (1 x)

We can see that

a = \frac{1}{4},

and

b = 1 .

Therefore, the graph of

g

is a vertical shrink by a factor of

\frac{1}{4}

of the graph of

f (x) = sin (x) .

Finding Period and Amplitude

Sketching the Graph

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

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

Exercise

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