How to Trace the Curve of (y sin^2 x) and (y 2sin^2 x)

When working with trigonometric functions, tracing curves can be a valuable tool for understanding their behavior. This article will guide you through the process of tracing the curves of (y sin^2 x) and (y 2sin^2 x), providing step-by-step instructions and insights for each function.

Step-by-Step Guide to Tracing (y sin^2 x)

Step 1: Understand the Function

The function (y sin^2 x) represents the square of the sine function, which oscillates between 0 and 1. Unlike the sine function, which has a period of (2pi), the function (y sin^2 x) has a period of (pi).

Step 2: Key Points

Identify key points within one period from (0) to (pi):

At (x 0): (y sin^2 0 0) At (x frac{pi}{2}): (y sin^2left(frac{pi}{2}right) 1) At (x pi): (y sin^2 pi 0)

Step 3: Symmetry

The function (y sin^2 x) is symmetric about (x frac{pi}{2}). Therefore, you can focus on the interval from (0) to (pi) and then mirror it for the interval (pi) to (2pi).

Step 4: Plot Points

Plot additional points to get a better shape of the curve:

At (x frac{pi}{4}): (y sin^2left(frac{pi}{4}right) left(frac{sqrt{2}}{2}right)^2 frac{1}{2}) At (x frac{3pi}{4}): (y sin^2left(frac{3pi}{4}right) frac{1}{2})

Step 5: Sketch the Curve

Start at ((0, 0)). Rise to (left(frac{pi}{2}, 1right)). Fall back to ((pi, 0)). Repeat this pattern for the next interval ([pi, 2pi]).

Step 6: Use a Graphing Tool

For better accuracy, you can use graphing software or a graphing calculator. Here’s a simple code snippet in Python using Matplotlib to plot the function:

import numpy as npimport  as plt# Define the x rangex  (0, 2*np.pi, 400)y  (x)**2# Create the plot(x, y, color'blue', lw0.5)plt.title('Graph of (y  sin^2 x)')plt.xlabel('x-axis')plt.ylabel('y-axis')plt.ylim(-0.1, 1.1)()

Tracing (y 2sin^2 x)

The function (y 2sin^2 x) is a scaled version of (y sin^2 x). Here’s how to trace this curve:

Step 1: Understanding the Function

The function (y 2sin^2 x) oscillates between 0 and 2. It has the same period as (y sin^2 x), which is (pi), but the amplitude is doubled.

Step 2: Key Points

Identify key points within one period from (0) to (pi):

At (x 0): (y 2sin^2 0 0) At (x frac{pi}{2}): (y 2sin^2left(frac{pi}{2}right) 2) At (x pi): (y 2sin^2 pi 0)

Step 3: Symmetry

The function (y 2sin^2 x) is symmetric about (x frac{pi}{2}). Therefore, you can focus on the interval from (0) to (pi) and then mirror it for the interval (pi) to (2pi).

Step 4: Plot Points

Plot additional points to get a better shape of the curve:

At (x frac{pi}{4}): (y 2sin^2left(frac{pi}{4}right) 2 times left(frac{sqrt{2}}{2}right)^2 1) At (x frac{3pi}{4}): (y 2sin^2left(frac{3pi}{4}right) 1)

Step 5: Sketch the Curve

Start at ((0, 0)). Rise to (left(frac{pi}{2}, 2right)). Fall back to ((pi, 0)). Repeat this pattern for the next interval ([pi, 2pi]).

Step 6: Use a Graphing Tool

For better accuracy, you can use graphing software or a graphing calculator. Here’s a simple code snippet in Python using Matplotlib to plot the function:

import numpy as npimport  as plt# Define the x rangex  (0, 2*np.pi, 400)y  2 * (x)**2# Create the plot(x, y, color'blue', lw0.5)plt.title('Graph of (y  2sin^2 x)')plt.xlabel('x-axis')plt.ylabel('y-axis')plt.ylim(-0.1, 2.1)()

Summary: The curve of (y sin^2 x) oscillates between 0 and 1 with a period of (pi). By plotting key points and using a graphing tool, you can visualize the complete behavior of the function. Similarly, (y 2sin^2 x) oscillates between 0 and 2, doubling the amplitude of (y sin^2 x).