Problem 57 - Algebra and Trigonometry

Question

Graph \(f\) and \(g\) on the same axes, and find their points of intersection. $$f(x)=3 \cos x+1, g(x)=\cos x-1$$

Step-by-Step Solution

Answer

The functions intersect at points where \(x = \pi + 2k\pi\).

Step 1: Understand the Functions

The given functions are: \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \). These are trigonometric functions that involve cosine.

Step 2: Find the Intersection Points Algebraically

Set the two functions equal to each other to find the points of intersection: \[3 \cos x + 1 = \cos x - 1\]Rearrange the equation to collect like terms: \[3 \cos x - \cos x = -1 - 1\]Simplify: \[2 \cos x = -2\]Divide by 2: \[\cos x = -1\]

Step 3: Solve for x

Recall that \( \cos x = -1 \) at specific angles. The general solution is \[x = \pi + 2k\pi\]for any integer \(k\).

Step 4: Analyze the Graph

Graph both \( f(x) = 3 \cos x + 1 \) and \( g(x) = \cos x - 1 \) on the same axes using graphing software or a calculator. Verify that the points \( x = \pi + 2k\pi \) are indeed where the functions intersect, which translates to points like \((\pi, -2), (3\pi, -2),\) etc.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry, often denoted as \( \cos x \). It measures the horizontal distance of a point on a unit circle from the center, starting from the x-axis. In simpler terms:

• The value of \( \cos x \) lies between -1 and 1.
• \( \cos 0 = 1 \) since at 0 degrees (or 0 radians), the point on the unit circle is at the farthest right.
• \( \cos \pi = -1 \) at 180 degrees, the point is at the farthest left of the circle.

Understanding the behavior of \( \cos x \) is essential when evaluating trigonometric functions, such as the ones given in the exercise. Each function, like \( f(x) = 3 \cos x + 1 \), shifts and stretches the basic cosine graph to form a new pattern.

Graphing Trigonometric Functions

To graph trigonometric functions like those in our exercise, you’ll need to consider:

• Amplitude: The height of the peaks from the center. For \( f(x) = 3 \cos x + 1 \), the amplitude is 3.
• Period: The length it takes for the function to repeat. Cosine functions typically have a period of \( 2\pi \).
• Vertical Shift: How much the function is moved up or down. Function \( f(x) \) is shifted up by 1 unit.

These transformations help us understand how the function behaves when graphed. Similarly, \( g(x) = \cos x - 1 \) has no stretch but moves the cosine function one unit downward. Graphing software or a graphing calculator can visually showcase these transformations, allowing you to pinpoint exact changes and intersections.

Intersection Points

Intersection points are where two functions share the same values for x and y coordinates. Mathematically, this happens when the equations of the functions are set equal to each other. In our example:\[3 \cos x + 1 = \cos x - 1\]Simplifying gives:\[2 \cos x = -2\]Which further resolves to \( \cos x = -1 \). This means the x-values where the cosine function has a value of -1 are the intersection points. By recognizing these points, particularly at \(x = \pi + 2k\pi\) for integer \(k\), you can precisely determine where the graphs of \(f(x)\) and \(g(x)\) intersect. This is critical for verifying intersections geometrically as well.

Solving Trigonometric Equations

Solving trigonometric equations is a process of finding angles that satisfy the equation. Typically:

• Identify the trigonometric function involved, in this case, cosine.
• Use algebraic manipulation to isolate the trigonometric part (e.g., \( \cos x = -1 \)).
• Determine the general solutions, which involve angle reference points known from the unit circle.

In our context, since \( \cos x = -1 \) occurs at \( x = \pi \), periodic properties of cosine allow us to write the general solution:\[x = \pi + 2k\pi\]Here, \(k\) represents any integer, covering all solutions given the periodicity of cosine. This complete understanding helps solve any trigonometric equation involving cosine, benefiting your broader math skills.