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Chapter 10: Problem 10

Sketch the graphs of the given functions. Check each using a calculator. \(y=35 \sin x\)

Short Answer

Expert verified
The graph of \( y = 35 \sin x \) has an amplitude of 35 and a period of \( 2\pi \). Check by a calculator for accuracy.

Step by step solution

01

Understand the Function

The given function is in the form of a sine function, specifically, it's a transformation of the basic sine function. The equation is \( y = 35 \sin x \). The coefficient "35" affects the amplitude of the sine wave.
02

Identify the Amplitude

The amplitude of a sine function \( y = a \sin x \) is the absolute value of "a". In this case, the amplitude is \( |35| = 35 \). This means the wave will oscillate 35 units above and below the x-axis.
03

Determine the Period

The period of the standard sine function \( y = \sin x \) is \( 2\pi \). Since there is no change in the period (the coefficient of \( x \) is 1), the period remains \( 2\pi \). This means the wave recurs every \( 2\pi \) units along the x-axis.
04

Sketch the Sine Curve

Draw the x and y axes on a graph. Plot the sine wave starting from the origin with a standard sine curve shape, but ensure that the peaks reach 35 and the troughs reach -35, with the wave completing one full cycle every \( 2\pi \) units on the x-axis.
05

Verify Using a Calculator

Use a graphing calculator to input the function \( y = 35 \sin x \). Check the graph displayed by the calculator to ensure it matches your sketch - look for consistent amplitude of 35 and a period of \( 2\pi \).

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Key Concepts

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

Sine Function
The sine function, typically written as \( y = \sin x \), is a fundamental trigonometric function that shows the relationship between the angles and sides of a right triangle.
  • It smoothly oscillates between -1 and 1, which serves as its basic range.
  • The function is periodic, meaning it repeats its values in regular intervals.
A sine curve starts at the origin and rises to 1, then falls down to -1 before returning to the origin. This creates the familiar 'wave' shape that repeats indefinitely.
  • It is often used to represent periodic phenomena like sound or light waves.
  • In its untransformed state, the sine function completes one full cycle every \( 2\pi \) units along the horizontal axis.
Amplitude
Amplitude is a crucial concept in understanding trigonometric functions like the sine function. It refers to the height of the wave from the center line to its peak.
  • Mathematically, for a sine function \( y = a \sin x \), the amplitude is given by \( |a| \).
For instance, in the function \( y = 35 \sin x \), the amplitude is 35. This implies that:
  • The wave stretches 35 units above and below the center line (which in the case of a sine function is the x-axis).
  • The amplitude directly influences the "height" of the peaks and "depth" of the troughs from the central horizontal line over which it oscillates.
This modification doesn't alter the horizontal stretch of the wave, but it makes it taller or shorter vertically.
Period of Trigonometric Functions
The period of a trigonometric function, such as the sine function, specifies the interval after which the function values repeat.
  • For the basic sine function \( y = \sin x \), the period is \( 2\pi \).
This means every \( 2\pi \) units along the x-axis, the sine function completes a full wave cycle, starting again from the origin.In functions like \( y = 35 \sin x \), unless there's a horizontal transformation of the sine function, the period remains unchanged at \( 2\pi \).
  • The introduction of any coefficient alongside \( x \), say \( y = \sin(bx) \), would alter this period. Specifically, the new period would be \( \frac{2\pi}{b} \).
Understanding the period helps in accurately sketching and predicting the behavior of trigonometric waves, which is vital in fields such as engineering and physics.

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Most popular questions from this chapter

Find the function and graph it for a function of the form \(y=a \cos x\) that passes through \((2 \pi,-2)\).

Use the trace feature of a calculator to find the value of a for either \(y=a \sin x\) or \(y=a \cos x,\) whichever is correct such that the given point is on the graph. The amplitude of each function is \(2.50 .\) Thereby determine the function. (All points are located such that the \(x\) value is between \(-\pi\) and \(\pi\).) $$(0.67,-1.55)$$

Parametric equations can be defined for functions other than trigonometric functions. In Exercises 33 and \(34,\) draw the graphs of the indicated parametric equations. In Exercises 35 and \(36,\) display the graphs on a calculator. $$x=4-t, y=t^{3}$$

Solve the given problems. Find the function and graph it for a function of the form \(y=2 \cos b x\) that passes through \((\pi, 0)\) and for which \(b\) has the smallest possible positive value.

Solve the given problems. In Exercises 41 and 42 use a calculator to view the indicated curves. The electric current \(i\) (in \(\mu \mathrm{A}\) ) in a certain circuit is given by \(i=3.8 \cos 2 \pi(t+0.20),\) where \(t\) is the time in seconds. Sketch three cycles of this function.
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