Chapter 6.6, Problem 39AYU

To determine

To find: Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis:

Answer to Problem 39AYU

Solution:

Sinusoidal functions produce graphs that look like waves, and any portion of the curve can be translated onto another portion of the curve. Circular motion and pendulum movement are excellent “real life” examples of sinusoidal relationships.

A sinusoidal function is a function that is like a $\text{sine}$ function in the sense that the function can be produced by shifting, stretching or compressing the $\text{sine}$ function.

You can graph sinusoidal functions using your knowledge of transformations.

There are 5 transformations that can affect trig functions:

• horizontal stretch (HS) alters the period
• vertical stretch (VS) alters the amplitude
• horizontal translation (HT) causes a phase shift relocating our initial point
• vertical translation (VT) relocates the sinusoidal axis
• reflection (± )changes the pattern of our 5 key points

Explanation of Solution

Sinusoidal functions produce graphs that look like waves, and any portion of the curve can be translated onto another portion of the curve. Circular motion and pendulum movement are excellent “real life” examples of sinusoidal relationships.

A sinusoidal function is a function that is like a $\text{sine}$ function in the sense that the function can be produced by shifting, stretching or compressing the $\text{sine}$ function.

You can graph sinusoidal functions using your knowledge of transformations.

There are 5 transformations that can affect trig functions:

• horizontal stretch (HS) alters the period
• vertical stretch (VS) alters the amplitude
• horizontal translation (HT) causes a phase shift relocating our initial point
• vertical translation (VT) relocates the sinusoidal axis
• reflection (± )changes the pattern of our 5 key points

Some basic vocabulary should be observed as shown in the diagram:

Period: a complete cycle of the graph where the graph starts to repeat.

Sinusoidal axis: horizontal line halfway between the local maximum and minimum.

Amplitude: vertical distance from the sinusoidal axis to the maximum or minimum

Local maximum: the tops of each “wave”, all equal in value

Local minimum: the bottoms of each “wave”, all equal in value

The relations involving $\text{sine}$ or $\text{cosine}$ can be expressed in the form:

$\frac{1}{A}(y-D)=\sin B(x-C)or\frac{1}{A}(y-D)=\cos B(x-C)$

• Determine the position of the sinusoidal axis.

  $\text{Sinusoidal axis = }\frac{\max +\min }{2}$

  This is the Vertical Translation or “D” in the formula.

• Determine the amplitude of the function.

  $\text{Amplitude = max – sinusoidal axis}$

  This is the Vertical Stretch or “A” in the formula.

• Determine the period of the graph for the function.

  $\text{Horizontal Stretch }=\frac{period}{360\text{º}}$

  This is the Horizontal Stretch or “B” in the formula.

• Determine the phase shift.

  $\text{Phase Shift = x }-\text{ coordinate of maximum}$

  This is the Horizontal Translation or “C” in the formula.

Graphing a Sinusoidal Function

The procedure for graphing sinusoidal functions is similar to quadratic functions.

$y=A\sin \left(\omega x-\varphi \right)+B$ or $y=A\cos \left(\omega x-\varphi \right)+B$

Here A is the amplitude function. We know the maximum and minimum value of $\text{sine}$ and $\text{cosine}$ is $\pm 1$. Therefore $y\text{-axis}$ takes value from $-A$ to $A$.

Both $\text{sine}$ and $\text{cosine}$ have the domain is $\left(-\infty ,\infty \right)$.

Therefore $x\text{-axis}$ takes values are $\left(-\infty ,\infty \right)$.