Problem 1 Determine the amplitude, the per... [FREE SOLUTION]

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Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 1

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using a graphing calculator. $$y=\sin x+1$$

Short Answer

Expert verified

Amplitude: 1, Period: 2\pi, Phase shift: 0. Shift the sine curve up by 1 unit.

Step by step solution

Identify the amplitude

The amplitude of a sine function in the form $y=a\sin(bx+c)+d$ is given by the absolute value of $a$. In this case, the function is $y=\sin x+1$, implying that $a=1$. Therefore, the amplitude is $\left|1\right|=1$.

Determine the period

The period of a sine function $y=a\sin(bx+c)+d$ is given by $\frac{2\pi}{b}$. Here, $b=1$. Therefore, the period is $\frac{2\pi}{1}=2\pi$.

Calculate the phase shift

The phase shift of a sine function $y=a\sin(bx+c)+d$ is determined by solving $bx+c=0$. In this function, $c=0$, so there is no phase shift.

Determine the vertical shift

The vertical shift of the function $y=a\sin(bx+c)+d$ is determined by $d$. In this case, $d=1$, so the whole graph is shifted up by 1 unit.

Sketch the graph

To sketch $y=\sin x+1$, start by plotting the basic sine curve $y=\sin x$. Then, shift the entire graph up by 1 unit. Points like $0, 1$, $\pi/2, 2$, $\pi, 1$, $3\pi/2, 0$, and $2\pi, 1$ can be plotted for accuracy.

Verify using a graphing calculator

After sketching the graph by hand, input $y=\sin x+1$ into a graphing calculator to verify the plotted graph matches the calculated shifts and shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude

The amplitude of a trigonometric function, specifically the sine function, refers to how much the graph oscillates above and below its central axis. For a function in the form $y = a\sin(bx + c) + d$, the amplitude is given by the absolute value of the coefficient $a$. In the case of the function $y = \sin(x) + 1$, the coefficient $a$ is 1. This means the amplitude is $ \left| 1 \right| = 1$. The graph will oscillate 1 unit above and below the central axis.

Period

The period of a sine function is the length of one complete cycle on the graph. For a general sine function $y = a\sin(bx + c) + d$, the period is calculated using the formula $\frac{2\pi}{b}$. For the function $y = \sin(x) + 1$, $b = 1$. Therefore, the period is $\frac{2\pi}{1} = 2\pi$. This means the function repeats its pattern every $2\pi$ units along the x-axis.

Phase Shift

The phase shift of a sine function determines how the graph is horizontally shifted from its usual position. For a sine function $y = a\sin(bx + c) + d$, the phase shift is calculated by solving $bx + c = 0$. In our function $y = \sin(x) + 1$, we have $c = 0$, which means there is no horizontal shift. Consequently, there is no change in the starting point of the sine wave on the x-axis due to phase shift.

Vertical Shift

The vertical shift of a sine function determines how the graph is shifted up or down along the y-axis. For the function $y = a\sin(bx + c) + d$, the vertical shift is given by $d$. In the function $y = \sin(x) + 1$, $d = 1$. This means that the entire graph of the sine function is shifted upward by 1 unit. Therefore, the midline of the graph moves from $y = 0$ to $y = 1$.

Sine Function

The sine function, $ \sin(x) $, is a fundamental trigonometric function characterized by its wave-like shape. It is periodic, meaning it repeats at regular intervals, and is symmetrical around its midline. For the standard sine function $y = \sin(x)$, the graph oscillates between -1 and 1, has a period of $2\pi$, and no phase or vertical shifts. In our modified function $y = \sin(x) + 1$, we see some changes:

The amplitude remains 1, as the coefficient of $\sin(x)$ is 1.
The period is still $2\pi$, meaning the pattern repeats every $2\pi$ units.
There is no phase shift since the function does not contain a horizontal translation component.
The vertical shift is 1, which shifts the graph upwards by 1 unit, moving the midline to $y = 1$.

To sketch this function, start by plotting $y = \sin(x)$ and then shift every point up by 1 unit. Important points to plot would include:

At $x = 0$, $y = 1$
At $x = \pi/2$, $y = 2$
At $x = \pi$, $y = 1$
At $x = 3\pi/2$, $y = 0$
At $x = 2\pi$, $y = 1$

This approach makes it easier to visualize and draw the graph manually.

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91影视

Determine the amplitude, the period, and the phase shift of the function and, without a graphing calculator, sketch the graph of the function by hand. Then check the graph using a graphing calculator. $$y=\sin x+1$$

Short Answer

Step by step solution

Identify the amplitude

Determine the period

Calculate the phase shift

Determine the vertical shift

Sketch the graph

Verify using a graphing calculator

Key Concepts

Amplitude

Period

Phase Shift

Vertical Shift

Sine Function

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