Chapter 6: Q. 22 (page 429)

In Problems 19–26, 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.
$y = \frac{1}{2} s e c (3 x - π)$

Short Answer

Expert verified

The graph of the function $y = \frac{1}{2} s e c (3 x - π)$ is:

Step by step solution

Step 1. Given

The function $y = \frac{1}{2} s e c (3 x - π)$

To graph the function with at least two cycles.

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

Compare the given function $y = \frac{1}{2} s e c (3 (x - \frac{π}{3}))$ with $y = A \sin (ω x - ϕ) + B$

No amplitude

Period, $\frac{2 π}{ω} = \frac{2 π}{3}$

Phase shift, $\frac{ϕ}{ω} = \frac{π}{3}$

Step 3. Determine x-coordinates

One cycle begins at $x = \frac{ϕ}{ω} (\frac{π}{3})$ and ends at

$x = \frac{ϕ}{ω} + \frac{2 π}{ω} = \frac{π}{3} + \frac{2 π}{3} = π$

To find the five key points, divide the interval $(\frac{π}{3}, π)$ into four sub intervals, each of length $\frac{2 π}{3} \div 4 = \frac{π}{6}$

$\frac{π}{3} + \frac{π}{6} = \frac{π}{2}$

$\frac{π}{2} + \frac{π}{6} = \frac{2 π}{3}$

$\frac{2 π}{3} + \frac{π}{6} = \frac{5 π}{6}$

$\frac{5 π}{6} + \frac{π}{6} = π$

The $x -$ coordinates are $\frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{5 π}{6}, π$

Step 4. Determine the key points

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

The key points include $(\frac{π}{3}, \frac{1}{2}), (- \frac{π}{3}, \frac{1}{2}), (\frac{2 π}{3}, - \frac{1}{2}), (- \frac{2 π}{3}, - \frac{1}{2}), (π, \frac{1}{2})$

Step 5. Sketch the graph

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

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In Problems 19–26, 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.
$y = \frac{1}{2} s e c (3 x - π)$

Short Answer

Step by step solution

Step 1. Given

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

Step 3. Determine x-coordinates

Step 4. Determine the key points

Step 5. Sketch the graph

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In Problems 19–26, 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=12sec(3x-π)

Short Answer

Step by step solution

Step 1. Given

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

Step 3. Determine x-coordinates

Step 4. Determine the key points

Step 5. Sketch the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 19–26, 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.
$y = \frac{1}{2} s e c (3 x - π)$