Problem 33 - Precalculus: A Concise Course

Question

Graph \(f\) and \(g\) on the same set of coordinate axes. (Include two full periods.) $$ \begin{array}{l} f(x)=\cos x \\ g(x)=2+\cos x \end{array} $$

Step-by-Step Solution

Answer

On a graph, \(f(x) = \cos x\) fluctuates from -1 to 1 over the period \(2\pi\), while \(g(x) = 2 + \cos x\) does the same between 1 and 3 (2 units higher).

Step 1: Identify Period

The cosine function has a period of \(2\pi\), so make sure to include two full periods, i.e., the x-axis should cover a range from \(0\) to \(4\pi\).

Step 2: Plot \(f(x) = \cos x\)

To start, plot the first function \(f(x) = \cos x\) as a wave that oscillates between \(-1\) and \(1\) over a period of \(2\pi\). Start at the point \((0,1)\), drop to \((\pi /2, 0)\) and \((\pi, -1)\), rise to \((3\pi /2, 0)\) and back to \((2\pi, 1)\). Repeat this one more time to cover two full periods.

Step 3: Plot \(g(x) = 2 + \cos x\)

Next, plot the second function \(g(x) = 2 + \cos x\). This is a simple vertical shift of the first function upwards by two units. To graph it, start at the point \((0,3)\), drop to \((\pi /2, 2)\) and \((\pi, 1)\), rise to \((3\pi /2, 2)\) and back to \((2\pi, 3)\). Repeat this one more time to cover two full periods.

Step 4: Resulting Graph

In the resulting graph, the function \(f(x)\) oscillates between \(-1\) and \(1\) while the function \(g(x)\) oscillates between \(1\) and \(3\). The peaks and troughs of function \(g(x)\) coincide with those of function \(f(x)\), but it is always two units higher because of the shift.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that appears frequently in mathematics and science. It is denoted by the symbol \(\cos\). If you image a circle centered at the origin of a coordinate system, the cosine function gives the x-coordinate of a point on the circle corresponding to a given angle. For a function of the form \(f(x) = \cos x\), the graph will create a wave-like pattern that oscillates above and below the x-axis.

For graphing the basic cosine function, remember these key points:

• The function’s highest value is 1 and the lowest value is -1.
• The wave starts at its highest point at \(x=0\), that is, at the coordinates \(0, 1\).
• The wave reaches the x-axis at \(\frac{\pi}{2}\) intervals, at points \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\), etc.
• At \(\pi\) and \(2\pi\), the function reaches its lowest and high points again, respectively.

When graphing, it’s important to get these key points accurately plotted before drawing the smooth, continuous wave of the cosine function.

Period of a Function

The period of a function refers to the interval after which the function’s values start to repeat. For trigonometric functions like the cosine function, this is the length of one complete cycle of the wave. The standard period of the cosine function is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the function will have completed a full cycle and will start repeating its values.

To graph two full periods of the cosine function \(f(x) = \cos x\), you would extend your graph to cover an x-range from 0 to \(4\pi\). This ensures that you see the wave pattern twice. The period of a function is a critical concept because it dictates how often the graph repeats itself. When graphing, closely paying attention to the period helps in creating an accurate representation of the function.

Vertical Shift

A vertical shift in a function occurs when every point of the function's graph is moved up or down by the same amount. This is what happens with the graph of \(g(x) = 2 + \cos(x)\) as compared to \(f(x) = \cos(x)\). A vertical shift does not change the shape of the function's graph; it merely slides it higher or lower.

For the exercise at hand, \(g(x)\) is the cosine function vertically shifted upwards by 2 units. To graph \(g(x)\) accurately, take every point on the \(f(x)\) graph and move it up by 2 units. The resulting graph will exhibit the same oscillating wave pattern as \(f(x)\), but it will be positioned above it, not crossing the x-axis but fluctuating between the values of 1 and 3, instead of -1 and 1. This concept is key when dealing with transformations of graphs, as it allows functions to be adjusted without altering their general form.