Problem 48 - Algebra 2

Question

Use a graphing calculator to graph each function in the interval from 0 to 2\(\pi .\) Then sketch each graph. $$ y=\sin x+2 x $$

Step-by-Step Solution

Answer

The resulting graph combines the regular sine wave with a linear function, causing it to increase upwards in the interval from \(0\) to \(2\pi \). Drawing the graph is based on the output from the graphing calculator.

Step 1: Input the function in the graphing calculator

The first step is to input the function \( y = \sin x + 2x \) into the graphing calculator. Be sure to set the viewing window of the calculator to include the range from \(0\) to \(2\pi \).

Step 2: Observing the output

The function should be graphed on the calculator. The shape of the function within the interval from \(0\) to \(2\pi \) is a combination of the regular sine wave and a linear function, increasing upwards.

Step 3: Sketching the graph

Finally, you are to use the graph displayed on the calculator as a guide and sketch an approximate representation of this on paper. Remember to mark the interval from \(0\) to \(2\pi \) on the x-axis, and corresponding points on the y-axis.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In particular, they are used primarily in the context of periodic phenomena. This includes oscillations and waveforms. The most common trigonometric functions are:

• Sine ( \(\sin\) )
• Cosine ( \(\cos\) )
• Tangent ( \(\tan\) )

The sine function, for example, produces a smooth wave-like curve when plotted. It starts at 0, rises to 1, falls to 0, descends to -1, and returns to 0 again through one complete cycle. This cycle repeats every \(2\pi\) in terms of the angle \(x\) measured in radians.In the exercise given, the function \(y = \sin x + 2x\) combines the properties of the sine wave with a linear function \(2x\). This results in a unique curve that captures the oscillatory nature of the sine wave and the steady incline of the linear component. Understanding how trigonometric functions work will allow you to predict and interpret their graphs effectively.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing functions including complex mathematical equations. These calculators can plot graphs, solve equations, and even analyze mathematical concepts.To graph a trigonometric function such as \(y = \sin x + 2x\), you'll need to:

• Turn on your graphing calculator.
• Navigate to the function input area.
• Enter the equation \(y = \sin x + 2x\) into the function editor (often labeled as "Y=" or similar).
• Set the viewing window to the range \(0\) to \(2\pi\) for accurate analysis.

This will help you visualize the function's behavior within the specified interval. The graphing calculator plots the sine curve's intricate oscillations superimposed with the linear growth, offering a clear representation of how these components interact. Reviewing this visual representation aids in better understanding the effect of each component of the function.

Function Sketching

Sketching a function by hand involves translating the graph observed on a graphing calculator into a manual drawing. It builds understanding of the function's behavior and aids in learning to work without technological aids.Here are steps for effectively sketching the function \(y = \sin x + 2x\):

• Start by marking your x-axis with intervals from \(0\) to \(2\pi\) and the y-axis with appropriate scale based on the graph's highest and lowest points.
• Identify key points: where the function starts ( \(x = 0\) , approximately at \(y = 0\) since \(\sin 0 = 0\) ) and ends ( \(x = 2\pi\) , with values based on the given formula).
• Note peaks, troughs, and intersection points: sine function's peaks and troughs and where the linear component affects height.
• Draw the curve smoothly between these points, illustrating both the oscillatory motion and the upward trend.

Sketching requires careful observation of the calculator's output and practice. While it might seem challenging initially, this improves your ability to estimate and predict function behaviors and shapes.