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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 6: Q. 21 (page 429)

In Problems 19鈥�26, apply the methods of this and the previous section to graph each function. Be sure to label key points and show at least two periods.
$y = 3 c s c (2 x - \frac{蟺}{4})$

Short Answer

Expert verified

The graph of the function $y = 3 c s c (2 x - \frac{蟺}{4})$ is:

Step by step solution

01

Step 1. Given

The function $y = 3 c s c (2 x - \frac{蟺}{4})$

To graph the function with at least two cycles.

02

Step 2. Find the amplitude, period and phase shift.

Compare the given function $y = 3 c s c (2 (x - \frac{蟺}{8}))$ with $y = A \sin (蝇 x - 蠒) + B$

No amplitude

Period, $\frac{2 蟺}{蝇} = 蟺$

Phase shift, $\frac{蠒}{蝇} = \frac{蟺}{8}$

03

Step 3. Determine x-coordinates

One cycle begins at $x = \frac{蠒}{蝇} (\frac{蟺}{8})$ and ends at

localid="1646463144043"

x = \frac{蠒}{蝇} + \frac{2 蟺}{蝇}

$= \frac{蟺}{8} + 蟺 = \frac{9 蟺}{8}$

To find the five key points, divide the interval $(\frac{蟺}{8}, \frac{9 蟺}{8})$ into four sub intervals, each of length $蟺梅 4 = \frac{蟺}{4}$

$\frac{蟺}{8} + \frac{蟺}{4} = \frac{3 蟺}{8}$

$\frac{3 蟺}{8} + \frac{蟺}{4} = \frac{5 蟺}{8}$

$\frac{5 蟺}{8} + \frac{蟺}{4} = \frac{7 蟺}{8}$

$\frac{7 蟺}{8} + \frac{蟺}{4} = \frac{9 蟺}{8}$

The $x -$ coordinates are $(\frac{蟺}{8}, \frac{3 蟺}{8}, \frac{5 蟺}{8}, \frac{7 蟺}{8}, \frac{9 蟺}{8})$

04

Step 4. Determine the key points

Use these values of $x$ to determine the key points on the graph:

The key points include $(\frac{- 5 蟺}{8}, 3), (\frac{3 蟺}{8}, 3), (- \frac{蟺}{8}, - 3), (\frac{7 蟺}{8}, - 3), (\frac{9 蟺}{8}, - 3)$

05

Step 5. Sketch the graph.

Plot these five points and fill in the graph of the function.

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Most popular questions from this chapter

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.

$\frac{2 蟺}{3}$

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.

$210 掳$

Use a calculator to find the approximate value of each expression rounded to two decimal places.

$\cot 70 掳$

Use the graph of the function to determine the domain and the range of function

$y = c s c (\frac{3 蟺}{2} x)$

Convert each angle in degrees to radians. Express your answer as a multiple of $蟺$ .

$30 掳$

See all solutions

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