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Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 4 - Graphs of the Circular FunctionsProblema 6
Capitolo 5, Problema 6

Fill in the blank(s) to correctly complete each sentence.
The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).

Guida verificata passo dopo passo
1
Identify the base function and the transformation: The base function is \(y = 2 \cos x\), and the given function is \(y = -5 + 2 \cos x\).
Recognize that adding or subtracting a constant outside the cosine function results in a vertical shift of the graph.
Since the function is \(y = 2 \cos x - 5\), this means the graph of \(y = 2 \cos x\) is shifted vertically by 5 units.
Determine the direction of the shift: because the constant is \(-5\), the graph shifts 5 units down.
Therefore, the graph of \(y = -5 + 2 \cos x\) is obtained by shifting the graph of \(y = 2 \cos x\) 5 units down.

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Vertical Shifts in Trigonometric Graphs

A vertical shift moves the entire graph of a function up or down without changing its shape. For y = 2 cos x, adding or subtracting a constant outside the cosine function shifts the graph vertically by that constant amount.
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Phase Shifts

Amplitude and Vertical Translation

The amplitude of y = 2 cos x is 2, which affects the height of peaks and troughs. Adding -5 shifts the graph down by 5 units, changing the midline from y=0 to y=-5, but the amplitude remains the same.
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Amplitude and Reflection of Sine and Cosine

Function Transformation Notation

In the function y = a cos x + d, 'd' represents the vertical shift. A positive 'd' shifts the graph up, while a negative 'd' shifts it down. Recognizing this helps identify how the graph moves relative to the base function y = a cos x.
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Introduction to Transformations
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