Problem 113 - Precalculus

Question

In Exercises \(111-114\), determine whether each statement is true or false. ( \(A\) and \(B\) are positive real numbers.) The graph of \(y=-A \cos (-B x)\) is the graph of \(y=A \cos B x\)

Step-by-Step Solution

Answer

The statement is false.

Step 1: Analyze the Negative Inside the Cosine Function

The function given is \(y = -A \cos(-B x)\). We know that cosine is an even function, which implies that \(\cos(-Bx) = \cos(Bx)\). So, the negative inside the cosine will not change the graph of the cosine function.

Step 2: Address the Negative Sign Outside the Function

In the expression \(y = -A \cos(-B x)\), the negative sign outside the cosine function \(-A\) implies a reflection of the cosine wave across the x-axis. This is different from \(y = A \cos B x\), which has a positive amplitude.

Step 3: Compare the Two Equations

Now, comparing both equations, the original equation is \(y = -A \cos(-B x)\), which reflects the graph of \(y = A \cos B x\) across the x-axis due to the negative amplitude. Therefore, they are not the same graph.

Key Concepts

Cosine Function

Cosine is one of the basic trigonometric functions, commonly known as a periodic function. It describes the ratio of the adjacent side to the hypotenuse in a right-angle triangle. Imagine the circle: when the cosine function is plotted, it creates a beautiful wave-like motion across the graph. This wave repeats every 360 degrees or \(2\pi\) radians. At the beginning of this wave, the value of cosine is at its peak, typically starting at 1 and moving down to -1 before repeating the cycle.

• Peak: When cosine has its maximum value.
• Trough: When cosine has its minimum value.
• Period: The length of one full cycle of the wave.

In our exercise, the given function is \(y = -A \cos(-B x)\). Notice how the negative signs might affect our understanding of the graph motion, but the nature of cosine remains a consistent, smooth oscillation.

Even Function Property

An even function is one that remains unchanged when its input is negated. Simply put, if you plug in the opposite of any input value, an even function will give you the same output as if you had plugged in the original number. Mathematically, this is expressed as \(f(-x) = f(x)\). Cosine is a perfect example of this property. For the cosine function, \(\cos(-Bx) = \cos(Bx)\). It doesn't matter if you stretch, compress, or even flip the input - as long as you're using cosine, this property holds. Because of this, in our original exercise, the negative inside \(\cos(-Bx)\) does not affect the overall output of the cosine function, leaving the graph of \(\cos(Bx)\) unaffected by the sign change.

Amplitude Reflection

Amplitude is the maximum height achieved by the wave. It can be visualized by the peaks and troughs of the curve for sine or cosine functions. In our task, the equation \(y = -A \cos(-B x)\) specifically has a negative sign in front of the amplitude, \(-A\). Here's where reflection comes into play. This negative sign flips the wave upside-down, performing a reflection across the x-axis.

• Positive Amplitude: Regular wave motion, starting at its peak.
• Negative Amplitude: Inverted wave, like holding a mirror under it.

Hence, the graph of \(y = -A \cos(-B x)\) ends up being a reflection of what would have been \(y = A \cos B x\). While the shape remains the same, the orientation is turned thereupon, that's why the claim in the exercise is false, revealing these functions as novel yet connected forms.