Question

Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility only to check your work. $$q(x)=3.6 \cos (\pi x / 24)+2$$

Short answer

Question: Sketch the graph of the function \(q(x) = 3.6 \cos (\frac{\pi}{24} x) + 2\) using its transformations and properties. Answer: To sketch the graph of the function \(q(x) = 3.6 \cos (\frac{\pi}{24} x) + 2\), we apply the following transformations to the base function \(y = \cos x\): 1. Vertical dilation: Stretch the graph vertically by a factor of 3.6 (amplitude becomes 3.6). 2. Horizontal dilation: Shrink the graph horizontally by a factor of \(\frac{\pi}{24}\) (period becomes 48). 3. Vertical shift: Shift the graph 2 units upwards. After applying these transformations, we get the graph of the function \(q(x) = 3.6 \cos (\frac{\pi}{24} x) + 2\). You can verify this by using a graphing utility to plot the given function.

Step-by-step solution

Identify the base function and transformations

The given function is \(q(x)=3.6 \cos (\pi x / 24) + 2\). The base function is \(y=\cos x\). The transformations of the graph involve vertical dilation, horizontal dilation, vertical shift, and a horizontal shift. Our goal is to determine the parameters of these transformations.

Vertical Dilation

A vertical dilation multiplies each \(y\)-value by a factor \(a\). In our case, \(a=3.6\). This stretches the graph by a factor of \(3.6\) in the vertical direction. The amplitude of the function becomes \(|3.6|\). The new function after the vertical dilation is: $$y = 3.6 \cos x$$

Horizontal Dilation

A horizontal dilation multiplies each \(x\)-value by a factor \(b\). In our case, \(b=\frac{\pi}{24}\). This shrinks the graph by a factor of \(\frac{\pi}{24}\) in the horizontal direction. The period of the base function is \(2\pi\). The new period after the horizontal dilation is: $$T=\frac{2\pi}{|b|} = \frac{2\pi}{\pi / 24} = 48$$ The new function after the horizontal dilation is: $$y = 3.6 \cos (\frac{\pi}{24} x)$$

Vertical and Horizontal Shifts

A vertical shift adds a value \(c\) to each \(y\)-value. In our case, \(c = 2\). This translates the graph by \(2\) units upwards. A horizontal shift moves the graph left or right by the value \(h\). In our case, there is no horizontal shift since there is no value added inside the cosine function's parentheses. The new function after both shifts is: $$y = 3.6 \cos (\frac{\pi}{24} x) + 2$$

Sketch the Graph

Now, we can sketch the graph by applying the transformations one by one: 1. Vertical dilation: Stretch the base graph (\(y=\cos x\)) vertically by a factor of \(3.6\). The graph will have an amplitude of \(3.6\). 2. Horizontal dilation: Shrink the graph horizontally by a factor of \(\frac{\pi}{24}\). The graph will have a period of \(48\). 3. Vertical shift: Shift the graph \(2\) units upwards. Sketching the graph with all the transformations applied, we can see the function represented as: $$q(x) = 3.6 \cos (\frac{\pi}{24} x) + 2$$

Verify with a Graphing Utility

Use a graphing utility, such as Desmos or GeoGebra, to plot the given function \(q(x) = 3.6 \cos (\frac{\pi}{24} x) + 2\). Verify that your graphical solution matches the graph generated by the graphing utility.

Key concepts

Graphing Trigonometric Functions

When you're graphing trigonometric functions like sine and cosine, understanding their shape is crucial. Both sine \(y = \sin x\) and cosine \(y = \cos x\) have wave-like patterns, repeating every \(2\pi\) units along the x-axis.
To graph these functions from scratch:

• Identify the base function (either \(y = \sin x\) or \(y = \cos x\)).
• Plot key points like the maximum, minimum, and intercepts.
• Sketch the basic wave shape between these points over one period.

For our function \(q(x) = 3.6 \cos \left( \frac{\pi}{24} x \right) + 2\), the base is the cosine function. The graph resembles a series of hills and valleys, with peaks representing maximum values and troughs representing minimum values. Before any transformations, the cosine function has specific points where it equals 1, 0, and -1 which help in mapping out these waves. If you're using a graphing utility, input the function and adjust the scale if necessary to better visualize these transformations.

Vertical Dilation

Vertical dilation involves stretching or compressing the graph vertically. This is controlled by the coefficient in front of the sine or cosine function. In the function \(q(x) = 3.6 \cos \left( \frac{\pi}{24} x \right) + 2\), the number 3.6 is the vertical dilation factor.
Here's what happens:

• Multiply each y-value of the original cosine graph by 3.6.
• This affects the graph’s amplitude, which is half the distance from the maximum to the minimum value of the function. Originally, the amplitude is 1, but with a dilation factor of 3.6, it becomes 3.6.

Vertical dilation doesn't affect the period (the distance required for one complete cycle of the wave). Instead, the graph’s peaks and troughs are stretched, making the waves more pronounced in the vertical direction. This shows clearly how significantly the graph extends above and below the horizontal axis.

Horizontal Dilation

Horizontal dilation changes the width of the waves in a trigonometric graph. It is determined by the constant inside the function’s argument, usually placed right before the variable x in the function. For the equation \(q(x) = 3.6 \cos \left( \frac{\pi}{24} x \right) + 2\), the term \( \frac{\pi}{24} \) shrinks the graph horizontally.
Here's how it works:

• The horizontal dilation affects the period of the cosine curve. Initially, our base cosine function completes one full wave or cycle in a distance of \(2\pi\).
• The period changes to \( \frac{2\pi}{(\pi/24)} = 48 \). This means each cycle of the cosine function now spans 48 units along the x-axis, making the waves more spread out.

Effectively, the graph becomes less frequent across the horizontal axis. This stretched appearance is a key transformation, as it significantly changes how the function’s behavior is visualized across varying inputs of x.

Vertical and Horizontal Shifts

Shifting the graph vertically or horizontally adjusts the position without altering its shape. Vertical shifts involve adding or subtracting a value outside of the function, while horizontal shifts occur inside the function's argument. For \(q(x) = 3.6 \cos \left( \frac{\pi}{24} x \right) + 2\), there are specific shifts:

• Vertical Shift: The "+2" outside the cosine function indicates a vertical shift upwards by 2 units. This moves the entire graph to sit higher on the y-axis than it would otherwise, effectively raising each point on the graph by 2 units.
• Horizontal Shift: In the given function, there’s no horizontal shift since there are no additional constants added or subtracted inside the cosine function's argument with x.

Shifts are an essential part of graph transformations. They help to align the graph according to specific requirements, making it match real-world scenarios better. Vertical shifts particularly alter the graph’s midline or axis of symmetry, which now becomes y = 2 instead of the original y = 0.