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Chapter 6: Problem 62

Use a graphing calculator to graph the function. $$y=7.5 \cos x+\sin 2 x$$

Short Answer

Expert verified
To graph \( y = 7.5 \, \cos(x) + \sin(2x) \), input the function into the graphing calculator and adjust the viewing window for clear visualization.

Step by step solution

01

Open the Graphing Calculator

Access the graphing calculator you are using. This could be a physical graphing calculator or an online graphing tool such as Desmos or GeoGebra.
02

Enter the Function

Input the given function into the graphing calculator. Ensure you enter it correctly as \( y = 7.5 \, \cos(x) + \sin(2x) \).
03

Adjust the Viewing Window

Set the viewing window to appropriately display the graph. For example, set the x-axis from \(-2\pi\) to \(2\pi\) and the y-axis to view the full range of the function, such as from -10 to 10.
04

Graph the Function

Press the graph button to display the function on the graphing calculator. Analyze the shape, peaks, and troughs of the graph.
05

Interpret the Graph

Observe key features of the graph, such as amplitude, frequency, and phase shifts associated with the cosine and sine components. Note where the function crosses the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing calculator
A graphing calculator is an essential tool for visualizing functions and understanding their behavior. These devices can plot graphs based on mathematical equations. To use a graphing calculator for the function \( y = 7.5 \, \text{cos}(x) + \text{sin}(2x) \):
  • Power on the calculator and navigate to the graphing mode.
  • Input the given function accurately. For multi-term functions, ensure each component is entered correctly.
  • Confirm the function by pressing 'enter' or the equivalent command based on your calculator model.
Graphing calculators, like Desmos or GeoGebra, often have user-friendly interfaces and tutorials. Utilizing these tools will give you a hands-on understanding of graph behavior and numerical changes.
cosine function
The cosine function, represented as \( \text{cos}(x) \), is a fundamental trigonometric function. Here are its main properties:
  • It has a periodic waveform with a period of \(2\pi\).
  • Its values range from -1 to 1.
  • It starts at 1 when \( x = 0 \).
In the given function \( y = 7.5 \, \text{cos}(x) + \text{sin}(2x) \), the term \(7.5 \, \text{cos}(x)\) indicates a cosine wave with a larger amplitude (7.5 times the usual amplitude). This will stretch the wave vertically, making its peaks and troughs more pronounced compared to the standard cosine wave.
sine function
The sine function, expressed as \( \text{sin}(x) \), is another core trigonometric function. Characteristics include:
  • It also has a periodic waveform with a period of \(2\pi\).
  • Its values range from -1 to 1.
  • It starts at 0 when \( x = 0 \).
In our equation, we see \( \text{sin}(2x) \). This indicates the sine wave is sped up, completing a full cycle twice as fast (its period is \( \frac{\text{2}\pi}{2} = \pi\)). Combined, \(7.5 \, \text{cos}(x) + \text{sin}(2x)\) forms a complex wave, combining both functions' properties.鈥� }, {

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Most popular questions from this chapter

Make a hand-drawn graph of the function. Then check your work using a graphing calculator. $$f(x)=2^{-x}$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y=x^{4}-x-2[4.1]$$

Find the signs of the six trigonometric function values for the given angles. $$-620^{\circ}$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=-\tan x$$

Find the \(x\) -intercept \((s)\) of the graph of the function. $$g(x)=x^{2}-x-6[3.2]$$
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