Problem 39 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=5 \sin x $$

Step-by-Step Solution

Answer

The graph of the function \(y = 5\sin x\) covers two full periods, has an amplitude of 5, a period of \(2\pi\), and no phase shift. The graph oscillates between -5 and 5, while repeating every \(2\pi\) units.

Step 1: Identify the Amplitude

The amplitude (the maximum absolute value the function can take) of a sine function is given by the absolute value of the coefficient of the variable inside the function. In this case, the amplitude is 5.

Step 2: Identify the Period and Phase Shift

The period of a sine function is the length it takes for the function to complete one cycle. For a standard sine function this is \(2\pi\). Since there is no coefficient of x within the sine function, the function has a period of \(2\pi\). Second, for a phase shift, if there is no addition or subtraction term inside the sine function, then there is no phase shift. So, this sine function has no phase shift.

Step 3: Sketch the Graph

Begin by drawing the basic shape of the sine function over one period, from 0 to \(2\pi\). Draw the points of interest such as the maximum, minimum, and intercepts at: (0,0), \((\pi/2, 5)\), \((\pi, 0)\), \((3\pi/2, -5)\), and \((2\pi, 0)\). Draw a smooth curve connecting these points. Repeat the shape for the second period from \(2\pi\) to \(4\pi\). Mark the y-axis at positive and negative 5 for the amplitude.

Key Concepts

Amplitude of Sine Function

The concept of amplitude in a sine function is vital as it determines the height of the wave from its midline. For the function \( y = 5 \sin x \), the amplitude is determined by the coefficient of the sine, which is 5.
This means that the sine wave will reach a maximum of 5 and a minimum of -5. In simpler terms, amplitude defines how tall and deep the waves of the graphed function will be.
Here's how to identify the amplitude:

• Look at the number directly in front of the sine function (in this example, 5).
• The absolute value of this number is the amplitude.

If you understand this, you're halfway there to understanding how the sine function graph behaves, specifically its range of oscillation.

Period of Sine Function

The period of a sine function defines how long it takes for the wave to start over or complete one full cycle. For the basic sine function \( y = \sin x \), this period is \(2\pi\).
In our given function \( y = 5\sin x \), since there is no coefficient altering \( x \), the period remains at \(2\pi\).
Here's a quick breakdown on identifying periods of sine functions:

• For a regular sine function, the expression is usually \( y = \sin(kx) \).
• The period is calculated by \( \frac{2\pi}{|k|} \).
• In our case, \( k \) is 1, so the period is \(\frac{2\pi}{1} = 2\pi\).

This understanding helps in knowing how frequently the wave repeats itself along the x-axis.

Sketching Graphs

Sketching the graph of the sine function \( y = 5 \sin x \) involves plotting it over two full periods as instructed.
The process can be broken down into a few key steps:

• Start by marking the crucial points over one period (0 to \(2\pi\)). These points include where the wave intersects the midline, and the maximum and minimum points which occur at multiples of \(\pi/2\).
• Specifically, you’ll place points at: \((0,0)\), \((\pi/2, 5)\), \((\pi, 0)\), \((3\pi/2, -5)\), and \((2\pi, 0)\).
• Connect these dots with a smooth, wave-like curve.
• Repeat this pattern from \(2\pi\) to \(4\pi\) to complete two full cycles.

Ensure your y-axis has marks at 5 and -5 to reflect the amplitude, providing a visual understanding of how the sine function varies. This approach makes graphing trigonometric functions more systematic and less intimidating.