Problem 69 - Algebra 2

Question

a Graph \(y=-\cos x\) and \(y=-\sec x\) on the same axes. b. State the domain, the range, and the period of each function. c. For which values of \(x\) does \(-\cos x=-\) see \(x ?\) Justify your answer. d. Writing Compare the two graphs. How are they alike? How are they different? e. Reasoning Is the value of \(-\) sec \(x\) positive when \(-\cos x\) is positive and negative when \(-\cos x\) is negative? Justify your answer.

Step-by-Step Solution

Answer

a. See the drawn graphs. b. The domain for both functions is all real numbers, the range for y=-cos x is -1 and 1, the range for y=-sec x is -Infinity, -1] U [1, Infinity, and the period is \(2\pi\) for both functions. c. \(-\cos x=-\sec x\) when \(x\) is an odd multiple of \(\pi /2\). d. Both graphs have the same period and domain, but the range and their values differ. e. No, the value of \(-\sec x\) is negative when \(-\cos x\) is positive and vice versa.

Step 1: Graphing the functions

First, graph the functions \(y=-\cos x\) and \(y=-\sec x\). Remember that the cosine function oscillates between -1 and 1, but the secant function, being the reciprocal function of cosine, has values everything else except within the range of (-1 , 1).

Step 2: State the domains, ranges, and periods

Next, state the domains, ranges, and periods for each function. The domain of both functions is all real numbers. The range for \(y=-\cos x\) is from -1 to 1 while the range for \(y=-\sec x\) tends to negative infinity, -1] union [1 to positive Infinity. The period for both functions is \(2\pi\).

Step 3: Determine when \(-\cos x=-\sec x\)

To answer when \(-\cos x = -\sec x\), remember that \(\sec x = 1/\cos x\). Therefore, these two functions are equal when \(\cos x = 1\) or \(-1\), which occurs when \(x\) is a multiple of \(\pi\). As for negative cos x and sec x, they are equal when \(x\) is an odd multiple of \(\pi /2\).

Step 4: Compare the two graphs

To compare the graphs, note the similarities and differences. Both graphs have the same period and domain, but their ranges and values differ. The -cos x graph oscillates while the -sec x graph has a hyperbolic shape with vertical asymptotes.

Step 5: Analyze the signs of \(-\cos x\) and \(-\sec x\)

Finally, to understand if the value of \(-\sec x\) is positive when \(-\cos x\) is positive and vice versa, remember that since sec x = 1/cos x, their signs will always be opposite. Therefore, when -cos x is negative, -sec x will be positive and vice versa.

Key Concepts

Domain and Range

When graphing trigonometric functions like \(y = -\cos x\) and \(y = -\sec x\), it's important to understand their domain and range.

The **domain** of a function refers to all the possible input values (\(x\)-values) that can be plugged into the function.

• For both the functions \(y = -\cos x\) and \(y = -\sec x\), the domain is all real numbers.

The **range** of a function is all the possible output values (\(y\)-values) that the function can produce.

• For \(y = -\cos x\), the range is from -1 to 1, as the cosine function oscillates between these values.
• For \(y = -\sec x\), which is the reciprocal of cosine, the range is every real number except the interval from -1 to 1. It includes values from \( -\infty\) to -1 and from 1 to \( \infty\).

Understanding the domain and range is crucial for graphing as it lets you know where to start and stop on the \(x\) and \(y\) axes.

Cosine Function

The cosine function, represented as \(y = \cos x\), is one of the fundamental trigonometric functions.
When its negative form, \(y = -\cos x\), is graphed:

• The curve reflects across the \(x\)-axis compared to the standard \(\cos x\) graph.
• It maintains a wave-like structure, oscillating between -1 and 1.
• Key points occur at multiples of \(\pi/2\), where it crosses the \(x\)-axis.

This function repeats its pattern every \(2\pi\) radians, which is known as its period.
Knowing the behavior and characteristics of the cosine function helps in predicting its transformations and plotting its graph.

Secant Function

The secant function is defined as the reciprocal of the cosine function, represented as \(y = \sec x = 1/\cos x\). When you graph \( y = -\sec x\):

• It has a series of vertical asymptotes where the cosine function is zero, at odd multiples of \(\pi/2\).
• The graph is hyperbolic in shape and diverges to \(\pm \infty\) around these asymptotes.
• A noteworthy feature is that it never intersects the interval [-1, 1]. Instead, it approaches negative values or spikes to positive infinity.

This function also repeats every \(2\pi\), aligning with how often the cosine zeros recur.
Understanding \(-\sec x\) contributes to grasping the behavior of reciprocal functions in trigonometric contexts.

Periodicity of Functions

Periodicity is a property of functions where they repeat their values at regular intervals.
Both the \(y = -\cos x\) and \(y = -\sec x\) functions exhibit periodicity, with a common period of \(2\pi\).

• The cosine function has a smooth and continuous wave that completes one full cycle by \(2\pi\).
• For the secant function, periodicity is characterized by its repeating sections of U-shaped curves and asymptotes. These occur because secant follows the cycles of cosine but swaps its zero intervals for infinite spikes.

The concept of periodicity is key in trigonometry as it allows you to predict function values and graph patterns over repeated intervals.
Recognizing this trait aids in understanding how trigonometric functions behave over extended ranges.