Problem 11 - Algebra and Trigonometry

Question

Graph the function. $$ g(x)=3+3 \cos x $$

Step-by-Step Solution

Answer

Graph \( g(x) = 3 + 3\cos x \) to oscillate between 0 and 6, with a period of \( 2\pi \).

Step 1: Understand the Parent Function

The parent function for this problem is the cosine function, which is represented as \( y = \cos x \). The graph of \( \cos x \) is a wave that starts at (0,1), peaks at \( y=1 \), bottoms at \( y=-1 \), and has a period of \( 2\pi \).

Step 2: Identify the Transformations

Our function is \( g(x) = 3 + 3 \cos x \). We notice there are two transformations: a vertical stretch and a vertical shift. The cosine function is being multiplied by 3 (a vertical stretch), and then 3 is added to the result (a vertical shift).

Step 3: Apply the Vertical Stretch

The vertical stretch is handled by multiplying the cosine function by 3. This means each y-value of the \( \cos x \) function is stretched three times farther from the x-axis. Thus, it oscillates between -3 and 3 instead of -1 and 1.

Step 4: Apply the Vertical Shift

The vertical shift is a transformation that moves the entire graph up by 3 units. Therefore, the new range of the function moves from \([-3, 3]\) to \([0, 6]\). Each point on the wave is moved 3 units up along the y-axis.

Step 5: Determine the Key Points to Plot

The key points of \( \cos x \) are at (0,1), \( (\pi/2, 0) \), \( (\pi, -1) \), \( (3\pi/2, 0) \), and \( (2\pi, 1) \). After applying the transformations, the key points for \( g(x) = 3 + 3\cos x \) become (0, 6), \( (\pi/2, 3) \), \( (\pi, 0) \), \( (3\pi/2, 3) \), and \( (2\pi, 6) \).

Step 6: Plot the Graph and Connect the Points

Using the key points determined in the previous step, plot these points on a coordinate grid. Then, connect these points to form the wave shape of the graph. Note that the graph has a maximum y-value of 6, a minimum y-value of 0, and retains the periodic nature of \( \cos x \) with a period of \( 2\pi \).

Key Concepts

Cosine Function

The cosine function is a foundational trigonometric function represented by the equation \( y = \cos x \). This function creates a smooth, continuous wave on the graph, known as a cosine wave. The basic graph of \( \cos x \) starts at the point (0, 1) on the y-axis.
It oscillates between 1 and -1, producing a pattern that repeats every \( 2\pi \) units along the x-axis. This property is called the function's period. The cosine wave is symmetric, reaching its highest point at 1 and its lowest at -1. Understanding this fundamental behavior is crucial when applying transformations such as stretching and shifting, as these will alter its appearance while maintaining certain characteristics like its periodic nature.
Key points to identify on the cosine graph include:

• Starting point: (0, 1)
• Top peak: y = 1
• Bottom peak: y = -1
• Period: \( 2\pi \)

Vertical Transformations

Vertical transformations involve modifications that change the y-values of a function's graph. They can include vertical stretches, compressions, and shifts, affecting the amplitude and position of the wave function along the y-axis.
In the given problem, the function \( g(x) = 3 + 3 \cos x \) involves two types of vertical transformations:

• **Vertical Stretch**: The original \( \cos x \) function is multiplied by 3, leading to a vertical stretch. This process extends the wave farther from the x-axis, so instead of oscillating between 1 and -1, it now oscillates between 3 and -3.
• **Vertical Shift**: Adding 3 to the equation shifts the entire wave graph upwards by 3 units. As a result, each point on the graph moves up three units, adjusting the range of the function from \([-3, 3]\) to \([0, 6]\).

These transformations modify the shape and positioning of the wave while the fundamental wave-like pattern of cosine remains intact.

Wave Graph

The wave graph of a trigonometric function like cosine depicts its characteristic oscillations. It showcases the repeating nature of these functions, with patterns that wave up and down regularly.
In the problem \( g(x) = 3 + 3 \cos x \), the changes render a new wave graph that is vertically stretched and shifted. This modified wave graph retains the periodic nature of the cosine, repeating every \( 2\pi \) units, but reaches a new maximum height of 6 and a minimum of 0 due to its transformations.
Some features of the wave graph to be aware of include:

• Peaks and troughs: The highest and lowest points of the wave.
• Crest to crest: The horizontal distance between two consecutive maximum points, defining the period.
• Smooth continuous form: The wave should pass through these key points smoothly without any abrupt changes.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals. Trigonometric functions such as sine and cosine are classic examples. They have periods, which define the length of one complete cycle on the graph before the pattern starts to repeat.
For the cosine function \( g(x) = 3 + 3 \cos x \), the period remains unchanged despite transformations. It continues to repeat every \( 2\pi \) units, aligning with the inherent properties of the cosine function. The resilience of the period, even after transformations, emphasizes a core trait of periodic functions: their cycle remains consistent.
Understanding periodic functions helps in graphing them accurately as well as in predicting their behavior over time:

• The period length determines the spacing between repeating waves.
• Key points repeat with every period, making plotting easier.
• Knowing the period allows predicting future values of the function.

Being aware of these periodic characteristics enhances the comprehension of trigonometric functions in both practical and theoretical applications.