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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 14
Chapter 1, Problem 14

Graph the functions in Exercises 13–22. What is the period of each function?


sin (x/2)

Verified step by step guidance
1
Identify the function: The function given is \( \sin(\frac{x}{2}) \). This is a sine function with a modified argument.
Determine the period of the sine function: The standard period of \( \sin(x) \) is \( 2\pi \). For \( \sin(\frac{x}{2}) \), the period is affected by the coefficient of \( x \) inside the sine function.
Calculate the new period: The period of \( \sin(kx) \) is \( \frac{2\pi}{|k|} \). Here, \( k = \frac{1}{2} \), so the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).
Graph the function: Start by plotting the key points of the sine function over one period \([0, 4\pi]\). These points include the intercepts, maximum, and minimum values.
Label the graph: Clearly indicate the period on the x-axis, and mark the amplitude (which remains 1 for this function) on the y-axis. The graph should show the wave completing one full cycle over the interval \([0, 4\pi]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Period of a Function

The period of a function is the length of the interval over which the function repeats itself. For trigonometric functions like sine and cosine, the period is a key characteristic that determines how often the wave pattern occurs. For example, the standard sine function, sin(x), has a period of 2π, meaning it repeats every 2π units along the x-axis.
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Graphs of Secant and Cosecant Functions

Transformation of Functions

Transformations involve changes to the basic form of a function, such as stretching, compressing, or shifting. In the case of sin(x/2), the 'x/2' indicates a horizontal stretch, which affects the period. Specifically, dividing the input by a factor (like 2) increases the period, resulting in a new period of 4π for sin(x/2).
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Graphing Trigonometric Functions

Graphing trigonometric functions requires understanding their key features, including amplitude, period, and phase shift. For sin(x/2), the graph will oscillate between -1 and 1, with a period of 4π, meaning it will complete one full cycle over this interval. Recognizing these features helps in accurately sketching the function and predicting its behavior.
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Related Practice
Textbook Question

Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.

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Textbook Question

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?


s = −tan πt

Textbook Question

Combining Functions


Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?


g. g ∘ f

Textbook Question

[Technology Exercise]


a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

Textbook Question

Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

Textbook Question

Graph the functions in Exercises 13–22. What is the period of each function?


sin (x − π/4) + 1