Problem 29 - Algebra 2

Question

Identify the period, range, and amplitude of each function. \(y=0.7 \cos \pi t\)

Step-by-Step Solution

Answer

The amplitude of the function \(y=0.7 \cos \pi t\) is 0.7, the period is 2, and the range is \([-0.7, 0.7]\).

Step 1: Identify the Amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of the function. In this case, the amplitude is given by \(|0.7| = 0.7\).

Step 2: Identify the Period

The period of a cosine function is given by \(\frac{2\pi}{|coefficient\ of\ t|}\). Here, the coefficient of \(t\) is \(\pi\) so the period is \(\frac{2\pi}{|\pi|} = 2\).

Step 3: Identify the Range

The range of a cosine function is given by \([-amplitude, amplitude]\). Since the amplitude is 0.7, the range is \([-0.7, 0.7]\).

Key Concepts

Amplitude

Amplitude is a key feature of trigonometric functions like sine and cosine. It determines the peak height of the wave from its midline. To find the amplitude of a function like \(y = 0.7 \cos \pi t\), we look at the coefficient of the cosine, which is 0.7 in this case.
This value tells us how far the peaks and troughs of the wave are from the center line. If the amplitude is a positive number, like 0.7, the wave peaks at 0.7 above the midline and troughs at 0.7 below it. Always, the amplitude is taken as the absolute value, which in this case is simply \(|0.7| = 0.7\).

• The amplitude is the half distance between the maximum and minimum values of the function.
• A larger amplitude means taller peaks and deeper troughs.
• Understanding amplitude helps visualize how intense or mild a wave is.

Period

The period of a function describes how long it takes for the function to complete one full cycle. For trigonometric functions like \(y = 0.7 \cos \pi t\), the period can be calculated using the formula \(\frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(t\) inside the function.
Here, \(b = \pi\). Thus, the period becomes \(\frac{2\pi}{|\pi|} = 2\). This means every 2 units along the horizontal axis (time in this scenario) the cosine wave repeats itself.

• Knowing the period helps predict when the function will perform regularly repeated behavior.
• A shorter period means the wave cycles more quickly, resulting in more waves over the same distance.
• For all cosine and sine functions, the "standard" period without any multiplier is \(2\pi\).

Range

The range of a function is the set of possible output values (y-values). For trigonometric functions like \(y = 0.7 \cos \pi t\), the range is determined by the amplitude since it governs the highest and lowest points of the wave.
Given that the amplitude is 0.7, the range of the cosine function is \([-0.7, 0.7]\). This indicates that the function oscillates between -0.7 and 0.7. The midline for the cosine function in this form is 0 since there’s no vertical shift added.

• The range of cosine functions without vertical shifting is determined as \([-\text{amplitude}, \text{amplitude}]\).
• The amplitude directly affects the maximum value of the range.
• Understanding the range is crucial for knowing the extent of the function's output.