How to Sketch the Graph of y sin 2x

To understand and sketch the graph of y sin 2x, we need to break down the process into several key steps. This is a detailed guide that will help you manage the visuals and properties of the sine function more effectively.

Understanding the Basics

The standard sine function, y sin x, has a period of 2π. For the function y sin 2x, the period is halved, becoming π. This indicates that the graph will oscillate twice as fast as the standard sine function.

Step 1: Identifying Key Points and Intervals

The basic period of the sine function is 2π. For y sin 2x, this period is reduced to π. We can start by dividing the x-axis into intervals of length π/2.

Step 2: Plotting Key Points

Key points for one period of y sin 2x are:

(0, 0) (π/4, 1) (π/2, 0) (3π/4, -1) (π, 0)

To plot these points, we start by noting that y sin 2(0) 0, y sin 2(π/4) sin(π/2) 1, y sin 2(π/2) sin(π) 0, y sin 2(3π/4) sin(3π/2) -1, and y sin 2(π) sin(2π) 0.

Step 3: Determining the Shape of the Graph

Once the key points are plotted, connect them smoothly to form the graph. The graph will have the same sinusoidal shape as the standard sine function but with more oscillations within one period. This means that it will have two peaks and two troughs within the interval from 0 to π.

Step 4: Extending the Graph

Since the period of the graph is π, we can extend the graph in both directions by repeating the same pattern of the key points. This will create a periodic function that repeats every π units.

Visual Example

Here is a basic sketch of the graph of y sin 2x for one period:

A basic representation of the graph of y sin 2x. Note that this sketch may not be to scale.

For a more detailed view, let's compare the standard sine function y sin x with y sin 2x. The graph of y sin x still in red and the graph of y sin 2x in blue.

The graph of y sin x (red) and y sin 2x (blue) graphed together. This visualizes how the latter function oscillates twice as fast as the former.

These visual aids are courtesy of Desmos, which is a powerful tool for learning and understanding mathematical concepts.