Problem 27 - Algebra 2

Question

State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. $$ y=6 \cos \theta+1.5 $$

Step-by-Step Solution

Answer

Vertical shift: 1.5 units up, Midline: \( y = 1.5 \), Amplitude: 6, Period: \( 2\pi \).

Step 1: Identify the vertical shift

The function given is \( y = 6 \cos \theta + 1.5 \). The vertical shift is determined by the constant added to the function, which is \( 1.5 \). This means the entire graph of the cosine function is shifted 1.5 units upwards.

Step 2: Identify the equation of the midline

The midline of the function is a horizontal line that represents the average value of the cosine function. For the given function, since it is vertically shifted by 1.5 units, the midline is \( y = 1.5 \). This is the line that passes through the average value of the modified cosine function.

Step 3: Determine the amplitude

The amplitude of a trigonometric function is the vertical distance from the midline to the maximum or minimum value of the function. The amplitude is represented by the coefficient of the cosine function, which is 6. Thus, the amplitude is 6.

Step 4: Calculate the period

The period of a cosine function is the distance required to complete one full cycle of the function. For the basic cosine function \( \cos(\theta) \), the period is \( 2\pi \). Since there are no modifications to the argument of the cosine, the period remains \( 2\pi \).

Step 5: Graph the function

To graph the function \( y = 6 \cos \theta + 1.5 \), start by drawing the midline at \( y = 1.5 \). The cosine curve will oscillate above and below this line with an amplitude of 6, reaching a maximum of \( 1.5 + 6 = 7.5 \) and a minimum of \( 1.5 - 6 = -4.5 \). The cycle repeats every \( 2\pi \) units. Plot these values and shape the smooth curve of the cosine function.

Key Concepts

Vertical Shift

The vertical shift of a trigonometric function represents the upward or downward movement of the graph from its original position. This shift is determined by a constant added to or subtracted from the function. In the equation \( y = 6 \cos \theta + 1.5 \), the vertical shift is the constant \( 1.5 \). This means that every point on the basic \( \cos \theta \) graph is moved 1.5 units upwards. Understanding vertical shifts is crucial because it helps in visualizing how the graph's position changes.

• A positive constant shifts the graph upwards.
• A negative constant shifts the graph downwards.

So, in this particular function, the entire graph of \( y = 6 \cos \theta \) is elevated by 1.5 units, allowing you to accurately place the function on a coordinate plane.

Amplitude

The amplitude of a trigonometric function like cosine or sine is a measure of how "tall" or "short" the graph is. It is the distance from the midline of the graph to its maximum or minimum points. In the function \( y = 6 \cos \theta + 1.5 \), the amplitude is 6. This value is derived from the coefficient of the cosine term.

• The amplitude affects the graph's range of height.
• It tells you how far the peaks and valleys of the graph move away from the midline.

For a clearer understanding:- If amplitude is 6, as in this function, it means the graph will reach 6 units above and below the midline. Hence, if the midline is at \( y = 1.5 \), the maximum point will be at \( y = 7.5 \), and the minimum at \( y = -4.5 \). This rhythmic up and down movement is essential for drawing the graph correctly.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. For the standard cosine and sine functions, this period is \( 2\pi \). In our function, \( y = 6 \cos \theta + 1.5 \), since there is no alteration in the angle (\( \theta \)), the period remains \( 2\pi \).

• The period helps us understand the repetition interval of the function.
• With each "period," the function returns to its beginning state.

To put it simply, if you were to trace the cosine curve starting from a point, it would take \( 2\pi \) units in \( \theta \) to return to the same point. Recognizing the period is crucial for accurately plotting repeated sections of the graph.

Midline

The midline of a trigonometric function is a horizontal line that serves as the baseline for the function's oscillations. In the equation \( y = 6 \cos \theta + 1.5 \), the midline is represented by \( y = 1.5 \). The function oscillates equally above and below this line.

• The midline can be thought of as the "average" value of the function.
• It shows the center of the function's "wave-like" pattern.

The importance of the midline lies in its role as a reference point. For this function:- Knowing the midline helps in determining the vertical positions of the maximum and minimum points, which correspond to the amplitude as discussed earlier. This clarity makes it easier to sketch the function and understand its behavior over a given domain.