Problem 25 - Contemporary Precalculus

Question

(a) State the rule of a function of the form $$f(t)=A \sin (b t+c)$$ whose graph appears to be identical with the given graph. (b) State the rule of a function of the form $$g(t)=A \cos (b t+c)$$ whose graph appears to be identical with the given graph. (Check your book to see graph)

Step-by-Step Solution

Answer

Answer: To find the functions \(f(t)\) and \(g(t)\) from a specified graph, follow these steps: 1. Identify the amplitude and period from the graph. 2. Determine the sine function, \(f(t) = A \sin(bt+c)\), using the amplitude and period. 3. Identify the phase shift for the sine function and determine the value of \(c\). 4. Determine the cosine function, \(g(t) = A \cos(bt+c)\), using the amplitude and period. 5. Identify the phase shift for the cosine function and determine the value of \(c\). By following these steps, you will obtain the functions \(f(t)\) and \(g(t)\) that match the given graph.

Step 1: Identify amplitude and period from the graph

Look at the given graph and identify the amplitude (\(A\)) and period. Amplitude is the maximum value of the function from the x-axis. The period can be determined by finding the length of one complete wave/cycle.

Step 2: Determine the sine function, \(f(t)\)

Using the amplitude and period identified in the previous step, we can find the sine function \(f(t) = A \sin(bt+c)\). Recall that the period of a sine function is given by \(\frac{2 \pi}{b}\). So, we can find the value of \(b\) using the identified period.

Step 3: Identify phase shift for sine function

Phase shift is the horizontal shift that the function has from its standard position. To find the phase shift, observe how much the sine function has been shifted horizontally and use it to determine the value of \(c\). The sine function \(f(t)\) becomes \(f(t) = A \sin(bt+c)\).

Step 4: Determine the cosine function, \(g(t)\)

Now, we need to find a cosine function that appears to be identical with the given graph. Since cosine functions have similar properties to sine functions, we start by using the same amplitude \(A\) and the period determined in step 1. The cosine function \(g(t)\) is given by \(g(t) = A \cos(bt+c)\).

Step 5: Identify phase shift for cosine function

Similar to step 3, observe the horizontal shift and determine the value of \(c\) for the cosine function \(g(t)\). The cosine function appears identical with the given graph and is given by \(g(t) = A \cos(bt+c)\). Without the actual graph, it's impossible to give exact values for \(A\), \(b\), and \(c\). However, the steps provided will help you determine the functions \(f(t)\) and \(g(t)\) when you analyze the graph from your book.

Key Concepts

Amplitude

The amplitude of a trigonometric function, such as sine or cosine, is a crucial concept because it indicates how much the function oscillates above and below its axis. It is the "height" of the wave. In more technical terms, the amplitude (\(A\)), is the maximum distance from the mean position (or the x-axis) to the peak or trough of the wave.

When analyzing the graph of a wave, look for the highest and lowest points. The amplitude is half the distance between these two extremes. If the peak is at a value of 3 and the trough at -3, the amplitude is:

• 3 - (-3) = 6 (total height)
• Amplitude = 6 / 2 = 3

Understanding amplitude is vital. It tells you how strong the oscillation is, whether you are dealing with sound waves, light waves, or electrical signals. The larger the amplitude, the stronger the wave. Hence, always check if the amplitude is properly identified for tackling trigonometric equations.

Period

The period of a trigonometric function determines how long it takes to complete one full cycle of its pattern. This can be visualized as the distance (in terms of the x-axis) required for the wave to repeat itself. Knowing the period allows you to predict where the function will next reach its peak, trough, or any other specific value.

For sine and cosine functions, the period is given by the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(t\) in the function \(f(t) = A \sin(bt+c)\) or \(g(t) = A \cos(bt+c)\). This means:

• If \(b = 1\), the period is \(2\pi\), which is the standard period for sine and cosine functions.
• When \(b > 1\), the period becomes shorter, resulting in more waves within the same x-axis length.
• Conversely, if \(b < 1\), the period is longer, meaning fewer waves over a given x-axis length.

Understanding the period helps in sketching graphs and solving equations, as it provides insight into the frequency of oscillations and the behavior of the wave over different intervals.

Phase Shift

Phase shift refers to the horizontal translation of the sine or cosine wave along the x-axis. It shows how the wave has been shifted left or right from the "standard" position. Understanding phase shift is essential when identifying the exact function that describes a given graph of a wave.

Consider the general form of the trigonometric function:\(f(t) = A \sin(bt+c)\) or \(g(t) = A \cos(bt+c)\). Here, \(c\) is the horizontal shift. The phase shift can be calculated as \(-\frac{c}{b}\). To determine it, first identify the original peak or support point, and then observe the horizontal shift of the graph.

• If \(c > 0\), the shift is towards the left.
• If \(c < 0\), the shift is towards the right.

Phase shift aligns the model with the actual data when you move or fit graph waves, ensuring accuracy. It modifies how a sine or cosine wave starts, crucial in both academic exercises and real-world applications like electrical engineering and signal processing.