Problem 21
Question
Describe the relationship between the graphs of \(f\) and \(g\). Consider amplitude, period, and shifts. $$ \begin{array}{l} f(x)=\cos 2 x \\ g(x)=-\cos 2 x \end{array} $$
Step-by-Step Solution
Verified Answer
The functions \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\) have the same amplitude of 1, the same period of \(\pi\), and no shifts. The difference lies in a reflection over the x-axis: \(g(x) = -\cos 2x\) is the reflection of \(f(x) = \cos 2x\) over the x-axis.
1Step 1: Identifying the amplitude
The amplitude of a trigonometric function is the absolute value of the coefficient of the function. For \(f(x) = \cos 2x\), the amplitude is 1 because the coefficient is 1. For \(g(x) = -\cos 2x\), the amplitude is also 1, because the absolute value of the coefficient -1 is 1.
2Step 2: Identifying the period
The period of a trigonometric function is found by \( \frac{2\pi}{|b|}\), where b is the coefficient of x. For both, \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), b = 2, so the period of both functions is \( \frac{2\pi}{2}\) or \(\pi\). This means both functions complete a cycle in interval of length \(\pi\).
3Step 3: Identifying shifts
If there's a constant being added or subtracted to the function, then there is a vertical shift; if it's inside the function with the independent variable, then there's a horizontal shift. In both \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), there is no additional constant either inside or outside, which means there are no shifts.
4Step 4: Comparing the graphs
The main difference between the two functions lies in the negative sign of \(g(x) = -\cos 2x\). This signifies a reflection on the x-axis. Hence, the graph of \(g(x)=- \cos 2x\) is the reflection of the graph of \(f(x)= \cos 2x\) over the x-axis.
Key Concepts
AmplitudePeriodReflection
Amplitude
In trigonometric functions, the amplitude refers to the height from the center line to the peak of the wave. It's essentially half the distance between the maximum and minimum values of the function. Amplitude is important because it tells you how "tall" or "short" the wave is.
In the context of our problem, both the functions \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\) have an amplitude of 1. This is derived from the absolute value of the coefficient of the cosine function, which is 1 in \(f(x)\) and \(-1\) in \(g(x)\).
The sign of the coefficient does not affect the amplitude. Therefore, both functions will reach the same maximum height above and below the center line of the graph despite one being positive and the other negative. This makes amplitude a critical parameter in analyzing the "intensity" or "strength" of the oscillation.
In the context of our problem, both the functions \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\) have an amplitude of 1. This is derived from the absolute value of the coefficient of the cosine function, which is 1 in \(f(x)\) and \(-1\) in \(g(x)\).
The sign of the coefficient does not affect the amplitude. Therefore, both functions will reach the same maximum height above and below the center line of the graph despite one being positive and the other negative. This makes amplitude a critical parameter in analyzing the "intensity" or "strength" of the oscillation.
Period
The period of a trigonometric function represents the interval needed for the function to complete one full cycle. This is a measure of how "stretched" or "squeezed" the cycles are along the x-axis. For the cosine and sine functions, computing the period involves figuring out how quickly the function repeats itself.
For any function of the form \(f(x) = \cos(bx)\), the period is determined using the formula \(\frac{2\pi}{|b|}\). In our scenario with both \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), the "b" value is 2. Hence, their periods are \(\frac{2\pi}{2} = \pi\).
This indicates that both functions repeat their cycle every \(\pi\) units along the horizontal x-axis. Regardless of transformations like amplitude changes or reflections, the period remains consistent because it's reliant on the coefficient of x in the cosine function.
For any function of the form \(f(x) = \cos(bx)\), the period is determined using the formula \(\frac{2\pi}{|b|}\). In our scenario with both \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), the "b" value is 2. Hence, their periods are \(\frac{2\pi}{2} = \pi\).
This indicates that both functions repeat their cycle every \(\pi\) units along the horizontal x-axis. Regardless of transformations like amplitude changes or reflections, the period remains consistent because it's reliant on the coefficient of x in the cosine function.
Reflection
A reflection in trigonometric functions involves flipping the graph over a certain axis, either horizontally or vertically. This transformation changes the orientation of the graph but not its amplitude or period.
In trigonometric functions, a negative sign in front of the function indicates a reflection over the x-axis. For function \(f(x) = \cos 2x\), it's simply a regular cosine wave. However, in the case of \(g(x) = -\cos 2x\), you have the same wave reflected over the x-axis due to the negative sign.
This means every peak or trough of \(f(x)\) becomes a trough or peak in \(g(x)\). The reflection affects the y-values of the function at each point—turning them from positive to negative or vice versa. Despite this reflection, both functions maintain identical amplitude and period, illustrating how reflections are distinct from other transformations like translations or dilations.
In trigonometric functions, a negative sign in front of the function indicates a reflection over the x-axis. For function \(f(x) = \cos 2x\), it's simply a regular cosine wave. However, in the case of \(g(x) = -\cos 2x\), you have the same wave reflected over the x-axis due to the negative sign.
This means every peak or trough of \(f(x)\) becomes a trough or peak in \(g(x)\). The reflection affects the y-values of the function at each point—turning them from positive to negative or vice versa. Despite this reflection, both functions maintain identical amplitude and period, illustrating how reflections are distinct from other transformations like translations or dilations.
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