Problem 20 - Contemporary Precalculus

Question

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=5 \cos t+3$$

Step-by-Step Solution

Answer

Answer: To transform the graph of \(f(t) = \cos t\) to the graph of \(g(t) = 5\cos t + 3\), the amplitude needs to be changed to \(5\), and a vertical shift of \(3\) units upwards should be applied.

Step 1: Determine the amplitude for both functions

Amplitude refers to the maximum value of a wave function from its mean value. For the given functions, the amplitude of the original function \(f(t) = \cos t\) is \(1\), as the maximum value of the cosine function is \(1\). The amplitude of the transformed function \(g(t) = 5\cos t + 3\) is \(5\), as the coefficient of the cosine function is \(5\).

Step 2: Determine the vertical shift for both functions

Now, we have to find the vertical shift, which refers to the position of the function on the vertical axis. The original function \(f(t) = \cos t\) has no vertical shift since it oscillates between \(-1\) and \(1\) with an average value of \(0\). The transformed function \(g(t) = 5\cos t + 3\) has a vertical shift of \(3\), because adding the constant \(3\) to the cosine function moves the entire graph upwards by \(3\) units.

Step 3: List down the required transformations

Based on the analysis, we can now list down the transformations needed to change the graph of the function \(f(t) = \cos t\) into the graph of the function \(g(t) = 5\cos t + 3\). 1. Change the amplitude of the original function from \(1\) to \(5\). This transformation is accomplished by multiplying the cosine function by \(5\). 2. Apply a vertical shift of \(3\) units upwards. This transformation is accomplished by adding the constant \(3\) to the transformed function. In conclusion, to transform the graph of \(f(t) = \cos t\) to the graph of \(g(t) = 5\cos t + 3\), the amplitude needs to be changed to \(5\), and a vertical shift of \(3\) units upwards should be applied.

Key Concepts

Amplitude of Trigonometric Functions

When you're working with trigonometric functions like the cosine and sine functions, the term 'amplitude' often comes up. In simple terms, the amplitude is a measure of the function's maximum change from its center position. To visualize this, imagine a wave: the amplitude would be the height of the wave's crest or the depth of its trough, measured from the calm, center line of the water.

For the basic cosine function, written as \(f(t) = \text{cos} t\), the amplitude is \(1\) because the function's values range from \(-1\) to \(1\), making the maximum displacement from the center line of the graph \(1\) unit. If you multiply the cosine function by a coefficient, say \(a\), as in \(g(t) = a \text{cos} t\), you've effectively changed the amplitude. Now, the range is from \(-a\) to \(a\), and this 'stretch' or 'shrink' in the wave corresponds directly with the amplitude.

Therefore, an important fact to remember is that the amplitude of \(f(t) = a \text{cos} t\) or \(f(t) = a \text{sin} t\) is the absolute value of \(a\), shown as \(|a|\).

Vertical Shift in Functions

Another transformation that can occur in functions is a vertical shift. This happens when you add or subtract a constant value from your function, effectively moving the graph up or down on the coordinate plane. Imagine your trigonometric wave is on a floating platform. If the platform rises or falls, the wave moves with it but keeps its shape. That's a vertical shift in a nutshell.

With the transformation of \(f(t) = \text{cos} t\) to \(g(t) = a \text{cos} t + k\), the \(+k\) at the end is your vertical shift. If \(k\) is positive, the shift is upward; if it's negative, the shift is downward. In the problem at hand, converting \(f(t) = \text{cos} t\) to \(g(t) = 5 \text{cos} t + 3\), the '+3' is what's responsible for the upward movement of the entire graph by three units along the vertical axis.

Cosine Function Transformations

Transformations of the cosine function can be intricate, as they can involve changes in amplitude, vertical and horizontal shifts, and even reflections. Understanding these changes requires you to examine the changes made to the function's formula.

In the context of the cosine function, multiplying the function by a coefficient, like in our exercise with \(g(t) = 5 \text{cos} t\), changes the amplitude. Adding or subtracting a number at the end of the function, as we have with the '+3' in \(g(t) = 5 \text{cos} t + 3\), results in a vertical shift.

Further tweaks can include horizontal shifts, achieved by adding or subtracting from the input variable \(t\), and reflections, which are caused by multiplying the entire function by \(-1\). Each of these adjustments transforms the classic cosine graph in a distinct way, giving you a wealth of possibilities for modeling real-world periodic behaviors, like the ebb and flow of tides or the oscillation of a pendulum.