Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=21+7 \sin (2 x+3) $$

Step-by-Step Solution

Answer

Amplitude: 7, Period: \(\pi\), Horizontal Shift: \(-\frac{3}{2}\) left, Vertical Shift: 21 up.

Step 1: Identify the standard form of sine function

The standard form for a sine function is \(y = A \sin(Bx + C) + D\). We need this form to identify the amplitude, period, and shifts of the function.

Step 2: Match your function to the standard form

Our function is \(y = 21 + 7 \sin(2x + 3)\). Here, we identify: \(A = 7\), \(B = 2\), \(C = 3\), and \(D = 21\).

Step 3: Determine the amplitude

The amplitude of the function is the absolute value of \(A\). Thus, amplitude \(= |7| = 7\).

Step 4: Calculate the period

The period of a sine function is given by \(\frac{2\pi}{B}\). For our function, \(B = 2\), so the period is \(\frac{2\pi}{2} = \pi\).

Step 5: Determine the phase shift

The horizontal shift (phase shift) is calculated by \(-\frac{C}{B}\). For our function, that's \(-\frac{3}{2}\), which means the graph is shifted to the left by \(\frac{3}{2}\) units.

Step 6: Determine the vertical shift

The vertical shift is given directly by \(D\). Thus, there is an upward shift of 21 units.

Step 7: Graph the function over the given interval

Plot the graph of the transformed sine function on the interval \(-5 \leq x \leq 5\), considering the amplitude of 7, a period of \(\pi\), a left horizontal shift of \(\frac{3}{2}\), and a vertical shift of 21. Start with the basic sine shape, adjust for periods and positioning, and make sure it's centered around \(y = 21\).

Key Concepts

Amplitude

The amplitude of a trigonometric function indicates how far the graph of the function stretches above and below its central axis. More simply, it shows the height of the wave from its central line to its peak. For the function \(y = 21 + 7 \sin(2x + 3)\), the amplitude is the absolute value of the coefficient of the sine function, known as \(A\).
In this case, \(A = 7\), which means the amplitude is \(\lvert 7 \rvert = 7\).
Thus, the graph of the function will rise 7 units above and fall 7 units below its midpoint. Remember, if \(A\) were negative, the amplitude would still be positive, as it is always an absolute value.

• The greater the amplitude, the "taller" the waves on the graph.
• An amplitude of 0 implies no wave movement, essentially a flat line along the vertical shift.

Period

A function's period is the length it takes for the function to repeat its pattern. In the context of trigonometric functions, this means the horizontal distance required for the graph to complete one full cycle.
For the function \(y = 21 + 7 \sin(2x + 3)\), the period is determined using the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = 2\), so the period is \(\frac{2\pi}{2} = \pi\).
What this means is the sine wave completes one full cycle per \(\pi\) units along the x-axis.

• A larger \(B\) will result in a more frequent repetition, thus shortening the period.
• Conversely, a smaller \(B\) will lengthen the period, spreading the wave further along the x-axis.

Phase Shift

Phase shift refers to the horizontal displacement of the graph from its usual position. In other words, it dictates where the function's cycle begins along the x-axis.
For the function \(y = 21 + 7 \sin(2x + 3)\), the phase shift is derived from \(-\frac{C}{B}\), where \(C\) is the constant added to \(x\) inside the sine function. Here, \(C = 3\), and \(B = 2\), giving us a phase shift of \(-\frac{3}{2}\).
This result indicates that the graph shifts \(\frac{3}{2}\) units to the left.

• If the shift is negative, the graph moves to the left, starting its cycle earlier.
• A positive shift pushes the graph to the right, delaying the start of the cycle.

Graphing Trigonometric Functions

Graphing a trigonometric function involves a stepwise process that incorporates the amplitude, period, and phase shift information to correctly place the function's wave on the coordinate plane. In this case, for the function \(y = 21 + 7 \sin(2x + 3)\), several key transformations are made to \(\sin(x)\) to result in the correct graph.
Start by recognizing the amplitude of 7, indicating that the wave will reach 7 units above and below the central axis, which in this equation is the vertical shift at \(y = 21\).
The period of \(\pi\) shows that one full oscillation cycle happens within a \(\pi\) distance on the x-axis, meaning the wave compresses compared to the standard \(2\pi\) period.
The phase shift moves the starting point \(\frac{3}{2}\) units to the left.
The vertical shift translates the entire graph upward by 21 units, centering it vertically around this new line.

• These transformations result in a wave that oscillates up and down centered around \(y = 21\).
• Always start by plotting points at critical values, such as the maximum, minimum, and intercepts, to assist in graphing.

By methodically mapping these attributes, one can effectively illustrate the function over any desired interval, such as \(-5 \leq x \leq 5\).