Problem 36 - Precalculus with Calculus Previews

Question

(a) sketch one cycle of the graph of the given function. (b) Find the amplitude \(|A|\) by inspection of the function. (c) Find the maximum value \(M\) and the minimum value \(m\) of the function on the interval in part (a). (d) Then use (4) to verify the amplitude \(|A|\) of the function. (e) Give the range of each function. $$ y=1-4 \sin \frac{2}{3} x $$

Step-by-Step Solution

Answer

Amplitude: 4; Maximum: -3; Minimum: 5; Range: [-3, 5].

Step 1: Understand the Function

The given function is a sine function in the form of \(y = C + A \sin(Bx)\), where \(C = 1\), \(A = -4\), and \(B = \frac{2}{3}\). This specific form allows us to identify the vertical shift, amplitude, period, and phase of the sine wave.

Step 2: Determine the Period

The period of a sine function \(y = A \sin(Bx)\) is given by \(\frac{2\pi}{|B|}\). For the given function \(B = \frac{2}{3}\), hence:\[\text{Period} = \frac{2\pi}{\left|\frac{2}{3}\right|} = \frac{2\pi}{2/3} = 3\pi\].This represents one full cycle of the sine wave.

Step 3: Sketch the Graph for One Cycle

To sketch \(y = 1 - 4 \sin\left(\frac{2}{3}x\right)\), begin by marking the period from \(0\) to \(3\pi\). At \(x = 0\), \(y = 1\); at \(x = \frac{3\pi}{2}\), \(y\) reaches its minimum, and at \(x = 3\pi\), \(y\) returns to the initial value. Plot these key points and draw a smooth curve.

Step 4: Identify the Amplitude by Inspection

The amplitude of \(y = 1 - 4 \sin\left(\frac{2}{3}x\right)\) is the absolute value of \(A\), which by inspection is \(|-4| = 4\).

Step 5: Find Maximum and Minimum Values

Since the function can be rewritten as \(y = 1 + (-4)(\sin(\frac{2}{3}x))\), we find that:- The maximum value of \(\sin(\frac{2}{3}x)\) is \(1\), giving the max value of \(y\) as \(1 - 4(1) = -3\).- The minimum value of \(\sin(\frac{2}{3}x)\) is \(-1\), giving the min value of \(y\) as \(1 - 4(-1) = 5\).

Step 6: Verify Amplitude Using Formula

Using the definition of amplitude \(|A| = \frac{M - m}{2}\), where \(M\) is the maximum and \(m\) is the minimum value:\[|A| = \frac{-3 - 5}{2} = 4\].This confirms the amplitude found by inspection.

Step 7: Determine the Range of the Function

The range is all values between the minimum and maximum inclusive. Therefore, the range of the function is \([-3, 5]\).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can be a fun and insightful way to understand how these functions operate. For the sine function given, which is in the form of \(y = 1 - 4 \sin \left(\frac{2}{3}x\right)\), graphing one cycle means showing how the function behaves from start to end, over one complete period.
To graph effectively:

• Identify the period, which in this case is \(3\pi\). This tells you how wide each cycle is on the x-axis.
• Mark key points, such as where the function starts (\(x=0, y=1\)), reaches its minimum (\(x=\frac{3\pi}{2}, y=-3\)), and comes back to start value (\(x=3\pi, y=1\)).

By connecting these points smoothly, you will have a clear graph depicting one cycle of the given sine function.

Amplitude of Trigonometric Functions

The amplitude of a sine function, like most trigonometric functions, tells us the height of the peaks or the depth of the troughs from the midline.
For the function \(y = 1 - 4 \sin \left(\frac{2}{3}x\right)\), the amplitude is given by the absolute value of the coefficient of the sine term \(A\), which is \(|-4| = 4\).

The concept here is straightforward but powerful—it measures how far the wave swings up and down from its central position. An amplitude of \(4\) means that the graph's crest is \(4\) units above the midline (which has been shifted to \(y=1\) due to the vertical shift). Understanding amplitude helps in predicting the high and low values the function can reach.

Period of Trigonometric Functions

The period of a trigonometric function indicates how long (or wide) it takes the function to complete a full cycle before repeating itself.
For the function \(y = 1 - 4 \sin \left(\frac{2}{3}x\right)\), the formula for period \(T\) is given by \(\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function.

Since \(B = \frac{2}{3}\), the period is: \[ T = \frac{2\pi}{\frac{2}{3}} = 3\pi \] This tells us that the full behavior of the sine wave—going up, down, and back to the base value—spans a horizontal width of \(3\pi\) on the x-axis. Recognizing the right period is essential while graphing, as it helps determine where to place the critical points on the graph.

Range of Functions

The range of a function refers to the set of all possible output values (y-values) the function can take. For the given function \(y = 1 - 4 \sin \left(\frac{2}{3}x\right)\), the range is determined by the amplitude and vertical shift.
Sine functions have outputs that originally vary from \(-1\) to \(1\). By applying the transformation in our function:

• The highest point occurs when \(\sin(\dots) = -1\), giving us a lower bound as \(1 - 4(-1) = 5\).
• The lowest point occurs when \(\sin(\dots) = 1\), resulting in \(1 - 4(1) = -3\).

Hence, the range is all values from \(-3\) to \(5\), written as \([-3, 5]\). Knowing what values the function can output is crucial for understanding its behavior and for solving related problems.