Problem 1 - Algebra and Trigonometry

Question

The amplitude \(A\) and period \(T\) of \(f(x)=5 \sin (4 x)\) are ______ and _______.

Step-by-Step Solution

Answer

The amplitude is 5 and the period is \(\frac{\pi}{2}\).

Step 1: Identify the Standard Form

Recall that the standard form of a sine function is given by \(f(x) = A \sin (B x)\). In this form, \(A\) represents the amplitude, and the period \(T\) is calculated using the formula \(T = \frac{2\pi}{B}\).

Step 2: Determine the Amplitude

In the given function, \(f(x) = 5 \sin (4 x)\), compare it to the standard form to identify \(A\). Here, \(A = 5\). Therefore, the amplitude is 5.

Step 3: Calculate the Period

Identify \(B\) from the given function, where \(B = 4\). Use the period formula \(T = \frac{2\pi}{B}\). Substituting \(B = 4\), the period \(T\) is \(T = \frac{2\pi}{4} = \frac{\pi}{2}\).

Key Concepts

Amplitude

The amplitude of a sine function is a measure of its peak value. It tells us how far the function's graph reaches from its middle value (usually zero) to its maximum or minimum value.
To find the amplitude, we look at the coefficient in front of the sine function, which is represented by the variable 'A' in the standard form:
\[f(x) = A \, \sin(Bx)\]
In the given function \(f(x) = 5 \, \sin(4x)\), the value of 'A' is 5. This means the function's amplitude is 5. The graph of this sine function will reach up to 5 and down to -5 from its central axis.

• Amplitude = peak value of the sine wave
• Found by identifying the coefficient A
• In the example, A = 5, so amplitude = 5

Period

The period of a sine function represents the length of one complete cycle of the wave. It tells us how far along the x-axis you go before the function begins to repeat. To find the period, we use the variable 'B' from the standard form and apply the formula:
\[T = \frac{2\pi}{B}\]
In the example function \(f(x) = 5 \, \sin(4x)\), 'B' is 4. Using the period formula, we get: \[T = \frac{2\pi}{4} = \frac{\pi}{2}\] This means the function completes one full cycle every \(\frac{\pi}{2}\) units along the x-axis.

• Period = length of one cycle
• Found by using the formula \(T = \frac{2\pi}{B}\)
• In the example, B = 4, so period = \(\frac{\pi}{2}\)

Standard Form

The standard form of a sine function provides a clear way to identify its amplitude and period. It is written as:
\[f(x) = A \, \sin(Bx)\]
Where:

• \(A\) is the amplitude
• \(B\) affects the period
• \(x\) is the input variable

The given function \(f(x) = 5 \, \sin(4x)\) matches this standard form. From this, we can easily determine the amplitude (\(A = 5\)) and calculate the period using \(B = 4\). Using the standard form helps us quickly understand the behavior of sine functions.

• The standard form is \(f(x) = A \, \sin(Bx)\)
• It helps identify amplitude and period easily
• In the example, \(A = 5\) and \(B = 4\), making it easy to find amplitude and period