Problem 22 - Algebra 2

Question

Identify the period, range, and amplitude of each function. \(y=3 \cos \theta\)

Step-by-Step Solution

Answer

The amplitude of the function \(y=3 \cos \theta\) is 3, the period is \(2\pi\), and the range is -3 ≤ y ≤ 3.

Step 1: Identifying the Amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of the \(\cos\) term. In the given function \(y=3 \cos \theta\), the coefficient of \(\cos \theta\) is 3. Thus, the amplitude of the function is |3| = 3.

Step 2: Identifying the Period

The period of a cosine function in its standard form \(y=a \cos(bx)\) is given by \(\frac{2\pi}{|b|}\). For the given function \(y=3 \cos \theta\), there is no coefficient next to \(\theta\), which means that the coefficient is 1. Therefore, the period of function is \(\frac{2\pi}{|1|}=2\pi\).

Step 3: Identifying the Range

The range of a standard cosine function is always between -1 and 1, inclusive. But here in our function \(y=3 \cos \theta\), the 3 in front of the cosine will stretch the standard range by a factor of 3. Therefore, the range of the function is -3 ≤ y ≤ 3.

Key Concepts

Amplitude

The amplitude of a trigonometric function, particularly cosine, reflects how "tall" or "short" the wave stretches vertically. It essentially measures the height of the wave from its central axis to the peak. In mathematical terms, the amplitude is the absolute value of the coefficient in front of the trigonometric function. For example, in the function \(y = 3\cos\theta\), the coefficient is 3, which means the wave will rise 3 units above and fall 3 units below its central axis.

• The amplitude is always positive because it represents a distance.
• For \(y = a\cos\theta\), the amplitude is \(|a|\).

Understanding amplitude helps predict how far a point on the function will move from its equilibrium position as it oscillates.

Period of a Function

The period of a trigonometric function is the horizontal length of one complete cycle of the wave. For cosine (or sine), this represents the distance until the function starts to repeat its shape.

• The standard period of \(\cos\theta\) is \(2\pi\), which is approximately 6.28 in decimal terms.
• In a function \(y = a\cos(bx)\), the period is calculated with \(\frac{2\pi}{|b|}\).

For example, in the function \(y = 3\cos\theta\), since there's no additional number multiplying \(\theta\), the period remains \(2\pi\). This means every \(2\pi\), the wave identically repeats itself starting a new cycle on the x-axis. Understanding the period is crucial for knowing how often a trigonometric function will repeat its pattern.

Range of a Function

The range of a function expresses the span of possible output values (y-values) that the function can produce. For a standard cosine function, the values oscillate between -1 and 1. However, when there is a coefficient multiplying the cosine function, it will scale this range.

• In a standard form function like \(y = a\cos\theta\), the range will extend from \(-a\) to \(a\).
• For \(y = 3\cos\theta\), the function output is scaled to rise as high as 3 and dip as low as -3.

Consequently, the range is \([-3, 3]\). This information helps determine the highest and lowest points the function can reach, essential for many applications, from engineering to physics.