Problem 39 - Precalculus : Building Concepts and Connections

Question

Graph the given pair of functions on the same set of axes. Are the graphs of \(f\) and \(g\) identical or not? $$f(x)=\cos (x+\pi) ; g(x)=\cos (x)+\cos \pi$$

Step-by-Step Solution

Answer

After graphing the two functions and comparing the graphs, we can see that the graphs of \(f\) and \(g\) are not identical. \(f(x)\) is a horizontal shift of the regular cosine function, while \(g(x)\) is a vertical shift of the cosine function.

Step 1: Understand the function

The cosine function has period \(2\pi\), which means that it repeats its values after each interval of \(2\pi\). The first equation \(f(x)=\cos (x+\pi)\), is a cosine function where \(x\) is shifted \(pi\) units to the left. The second equation \(g(x)=\cos (x)+\cos \pi\), is a sum of a regular cosine function and the cosine of \(\pi\), which is constant and equals -1.

Step 2: Graph the functions

Graph the functions \(f(x)=\cos (x+\pi)\) and \(g(x)=\cos (x)+\cos \pi\). Observe that the first function is only shifted horizontally on the graph, while the second function is shifted vertically because the cosine of \(\pi\) is subtracted.

Step 3: Compare the graphs

After plotting both functions, compare the graphs. If the shape, position, and direction of the two graphs are identical, the functions are identical.

Key Concepts

Cosine Function Period

The period of a cosine function refers to the interval after which the values of the function begin to repeat. For the basic cosine function, denoted as \(f(x) = \cos(x)\), this period is \(2\pi\). Why is that so? Because cosine is a trigonometric function that models wave-like patterns, such as sound waves or light waves, and the period corresponds to a full cycle of the wave.

For any cosine function, the period can be determined by observing how often the pattern repeats along the x-axis. If a cosine function is described as \(f(x) = \cos(bx)\), the period is adjusted to \(\frac{2\pi}{|b|}\), where \(b\) is a real number that stretches or compresses the graph horizontally. This concept is crucial to understand before graphing cosine functions, as it allows us to predict where the function will repeat its behavior across the x-axis.

For the exercise in question, the equation \(f(x) = \cos(x + \pi)\) adheres to the basic cosine function's period; hence, regardless of the horizontal shift, we know that the function will complete a full cycle every \(2\pi\) units.

Horizontal Shifts in Trigonometric Functions

Horizontal shifts in trigonometric functions occur when the input variable, \(x\), is replaced with \(x - h\) or \(x + h\), where \(h\) represents the number of units that the function is shifted horizontally on the coordinate plane. A positive \(h\) in \(x + h\) indicates a shift to the left, while a negative \(h\) in \(x - h\) represents a shift to the right.

When graphing, these shifts can profoundly affect the appearance of the function's graph, though they don't alter the shape or period of the wave pattern. For example, the function \(f(x) = \cos(x + \pi)\) in the exercise involves a shift of '\(\pi\)' units to the left because of the addition of \(\pi\) inside the cosine function's argument. This understanding is critical in identifying and comparing the features of trigonometric graphs.

Identifying Horizontal Shifts

When observing a graph, a horizontal shift manifests as the entire function moving along the x-axis without changing the amplitude or wave frequency. In our current exercise, recognizing this shift helps us to anticipate where the crests and troughs of the cosine curve will be located after the transformation is applied.

Sum of Trigonometric Functions

The sum of trigonometric functions occurs when two or more trig functions are added or subtracted from each other. The resulting function reflects a combination of the effects of each individual function. When dealing with sums, it is essential to remember that each function operates independently before their effects are combined.

For the given exercise with \(g(x) = \cos(x) + \cos(\pi)\), the sum is slightly different because \(\cos(\pi)\) is not a function but a constant, with a value of '-1'. In essence, we are adding a constant to the value of \(\cos(x)\) at each point. This means that every point on the graph of \(\cos(x)\) will be translated exactly one unit downwards, creating a new function entirely.

Visualizing Sum Effects

To visualize such sums, one can consider graphing each trigonometric function separately and then combining the graphs' corresponding y-values. This visual aid can highlight the blended effect on the resulting graph's amplitude and vertical shift, which is particularly helpful when the sum involves more complex or numerous functions.