Problem 34 - 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. $$ y=-1-5 \cos \pi x $$

Step-by-Step Solution

Answer

Amplitude is 5; maximum value is 4; minimum value is -6; range is [-6, 4].

Step 1: Identify the Form of the Function

The function given is \( y = -1 - 5 \cos(\pi x) \). This is in the form of \( y = C + A \cos(Bx) \), where \( A = -5 \), \( B = \pi \), and \( C = -1 \).

Step 2: Sketch One Cycle of the Graph

To sketch one cycle of \( y = -1 - 5\cos(\pi x) \), determine the period. The period of a cosine function \( A \cos(Bx) \) is given by \( \frac{2\pi}{B} \). Thus, for this function, it is \( \frac{2\pi}{\pi} = 2 \). The cycle repeats every 2 units along the x-axis.

Step 3: Find the Amplitude

By inspection of the function \( y = -1 - 5 \cos(\pi x) \), the amplitude \(|A|\) is equal to the absolute value of the coefficient of \( \cos \), which is \(|-5| = 5\).

Step 4: Find Maximum and Minimum Values

Since \( y = C + A\cos(\pi x) \), the maximum value is \( C + |A| = -1 + 5 = 4 \) and the minimum value is \( C - |A| = -1 - 5 = -6 \). Thus, the maximum value on the interval is 4 and the minimum value is -6.

Step 5: Verify the Amplitude Using Definition

Verify amplitude \(|A|\) using formula \((M - m)/2\), where \( M \) is the maximum and \( m \) is the minimum. Therefore, the calculation is \((4 - (-6))/2 = 10/2 = 5\), confirming that the amplitude is 5.

Step 6: Determine the Range of the Function

The range of the function \( y = -1 - 5 \cos(\pi x) \) is from the minimum value to the maximum value, which is \([-6, 4]\).

Key Concepts

Amplitude

The amplitude of a cosine function, such as in the equation \( y = -1 - 5\cos(\pi x) \), represents half the distance between its maximum and minimum values. It's a measure of how "tall" the wave's peaks are. In mathematical terms, amplitude is the absolute value of the coefficient in front of the cosine term. Here, that's \(-5\), so the amplitude is \(|A| = |-5| = 5\). This means that the graph of this function will reach 5 units above and below its midline, which is determined by the \( C \) value in the formula \( y = C + A\cos(Bx) \). In this case, the midline is at \(-1\).

• The amplitude is always a positive value.
• It affects the vertical stretching or compressing of the graph.
• The amplitude doesn't change the period of the function.

With this understanding, remember that the amplitude only influences the vertical rise and fall of the curve, not its width along the x-axis.

Period

In the context of trigonometric functions like cosine, the period is the horizontal length required for a function to complete one full cycle. For the function \( y = -1 - 5\cos(\pi x) \), the period can be determined by the formula \( \frac{2\pi}{B} \), where \( B \) is the coefficient multiplying the variable \( x \) in the cosine term. Here, \( B = \pi \), hence the period is \( \frac{2\pi}{\pi} = 2 \).

• The period tells us how often the wave pattern repeats itself.
• A smaller \( B \) value increases the period (stretches the graph horizontally).
• Larger \( B \) values decrease the period (compresses the graph horizontally).

This means that every 2 units along the x-axis, the wave for \( y = -1 - 5\cos(\pi x) \) completes one entire cycle of repeating its shape. Recognizing the period is crucial for graphing trigonometric functions accurately.

Maximum Value

The maximum value of a cosine function is significant as it shows the highest point that the graph reaches. In the function \( y = -1 - 5\cos(\pi x) \), the maximum can be calculated using \( C + |A| \). Here, \( C = -1 \) and the amplitude is \( |A| = 5 \), so the maximum value of the function is \( C + |A| = -1 + 5 = 4 \).

• Maximum value is reached when \( \cos(Bx) = 1 \).
• This value tells us the upper boundary of the function's range.
• In terms of graphing, it defines the peak points.

So, for this function, the highest point on the graph over any given period occurs at 4. This knowledge is valuable when determining the range of output values the function can produce.

Minimum Value

The minimum value of a cosine function reflects the lowest point reached by the graph. For the cosine function \( y = -1 - 5\cos(\pi x) \), we determine this using \( C - |A| \). With the values \( C = -1 \) and amplitude \( |A| = 5 \), the minimum is found to be \( C - |A| = -1 - 5 = -6 \).

• Minimum value is reached when \( \cos(Bx) = -1 \).
• This value sets the lower boundary for the function's output range.
• Knowing this helps in sketching the troughs of the graph.

As a result, the lowest point the graph reaches is -6. Understanding both maximum and minimum values allows for a complete understanding of the range of the function, helping to graph it correctly.