Question

Graphing sine and cosine functions 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 to check your work. $$q(x)=3.6 \cos (\pi x / 24)+2$$

Short answer

Answer: The amplitude is 3.6, the period is 48, the phase shift is 0, and the vertical shift is 2.

Step-by-step solution

Identify Amplitude

In \(q(x) = 3.6 \cos(\frac{\pi x}{24}) + 2,\) the amplitude can be found by looking at the coefficient of the cosine function, which is 3.6. Therefore, the amplitude is \(|3.6|=3.6\).

Identify the Period

The period can be found by looking at the coefficient of the \(x\) in the cosine function. We have \(\frac{\pi}{24}\) as the coefficient of \(x\). The period formula for cosine functions is given as: $$T = \frac{2\pi}{k}$$ In our case, \(k = \frac{\pi}{24}\). By plugging this \(k\) into the formula, we have: $$T = \frac{2\pi}{(\frac{\pi}{24})} = 2\pi \cdot \frac{24}{\pi} = 48$$ The period of the function is 48.

Identify Phase Shift

In our function \(q(x) = 3.6 \cos(\frac{\pi x}{24}) + 2\), there is no phase shift since there is no additional constant added or subtracted inside the cosine function. Therefore, the phase shift is 0.

Identify Vertical Shift

The vertical shift can be found by looking at the constant term in the function. In our case, the vertical shift is +2.

Sketching the graph

To sketch the graph, follow these guidelines: 1. Draw the midline at y=2 (due to the vertical shift). 2. Mark the amplitude of 3.6 above and below the midline. 3. Divide the x-axis into intervals equal to the period (48). 4. Draw the cosine wave fluctuating between the amplitude and the midline.

Check with a graphing utility

Use a graphing calculator or an online graphing tool to verify if the graph is correct. Compare the graph obtained with the sketched graph.

Key concepts

Sine and Cosine Graph Transformations

When you're dealing with sine and cosine functions, a few transformations can significantly alter the appearance of their graphs. These transformations include shifts (horizontal or vertical) and scaling (stretching or compressing), which modify the wave-like nature of these trigonometric functions. For the given function, \[ q(x) = 3.6 \cos\left(\frac{\pi x}{24}\right) + 2 \]we focus on how the standard cosine function, \(y = \cos x\), transforms to this new form.

• First, observe the scaling effect due to the coefficient 3.6. This is the factor that stretches the graph vertically, which defines the amplitude.
• Then, consider any shifts: the addition of a constant outside the cosine function indicates a vertical shift. In this case, the entire function shifts up by 2 units, thus moving the midline from \(y=0\) to \(y=2\).
• Lastly, the fraction \(\frac{\pi}{24}\) in the argument of the cosine affects the period, determining the horizontal stretch or compression of the wave.

Understanding these transformations will help in sketching and predicting how the graph will appear without even plotting point by point.

Amplitude and Period in Trigonometry

Amplitude and period are key concepts in understanding the oscillatory behavior of trigonometric functions like sine and cosine. The amplitude is derived from the coefficient in front of the sine or cosine term, and it indicates how far the graph will extend above and below its midline (central axis).

• For the function \(q(x) = 3.6 \cos(\frac{\pi x}{24}) + 2\), the amplitude is straightforwardly the absolute value of 3.6, meaning the wave reaches 3.6 units above and below the midline.

The period is less intuitive but no less important. It represents the horizontal length over which the function completes one full cycle before repeating. Using the period formula,\[ T = \frac{2\pi}{k} \]for a cosine function, where \(k\) is the coefficient of \(x\), we determined \(T=48\), showing that the graph repeats every 48 units. This means that if you plotted the function, you'd see it complete a single wave after moving 48 units along the x-axis.

Graphing Utilities in Mathematics

Graphing utilities offer a remarkable way to visualize mathematical functions, making abstract concepts more tangible. They come in the form of calculators or online tools and act as excellent educational aids, confirming whether hand-sketched graphs align with actual function behavior. For our function, using a graphing utility provides several benefits:

• First, they allow you to validate theoretical transformations applied, like amplitude adjustments and period calculations. You can instantly view how accurate your understanding of these transformations is by comparing it to the tool's output.
• These utilities also help in exploring the effects of different parameter changes without manual recalculations, offering insight into how functions behave beyond simple sketches.

Also, by visualizing the graph from a utility, you can spot any potential errors in your manual calculations, acting as a cross-verification for consistency in your work. Thus, they are invaluable resources for both learners and seasoned mathematicians alike.