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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 21
Capitolo 1, Problema 21

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


sin (x − π/4) + 1

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Identify the function to be graphed: \( f(x) = \sin(x - \frac{\pi}{4}) + 1 \). This is a transformation of the basic sine function.
Determine the horizontal shift: The term \( x - \frac{\pi}{4} \) indicates a phase shift to the right by \( \frac{\pi}{4} \) units.
Determine the vertical shift: The '+1' outside the sine function indicates a vertical shift upwards by 1 unit.
Identify the period of the function: The period of the basic sine function \( \sin(x) \) is \( 2\pi \). Since there is no coefficient affecting the \( x \) inside the sine function, the period remains \( 2\pi \).
Graph the function: Start by plotting the basic sine curve, apply the horizontal shift to the right by \( \frac{\pi}{4} \), and then shift the entire graph upwards by 1 unit. The resulting graph will have the same shape as the sine curve but will be shifted accordingly.

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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 function cycles through its values. For the sine function, the standard period is 2π, meaning it completes one full cycle over this interval.
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Graphs of Secant and Cosecant Functions

Transformation of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. In the given function sin(x − π/4) + 1, the term (x − π/4) indicates a horizontal shift to the right by π/4 units, while the +1 indicates a vertical shift upward by 1 unit. Understanding these transformations is essential for accurately graphing the function.
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Intro to Transformations

Graphing Trigonometric Functions

Graphing trigonometric functions requires knowledge of their basic shapes and how transformations affect these shapes. The sine function typically oscillates between -1 and 1, and when transformed, its amplitude and vertical position can change. For sin(x − π/4) + 1, the graph will oscillate between 0 and 2, reflecting the vertical shift.
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Introduction to Trigonometric Functions
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