Problem 43 - Algebra 2

Question

Which equation has the same graph as \(y=-\cos t ?\) F. \(y=\cos (-t) \quad\) G. \(y=\sin (t-\pi) \quad\) H. \(y=\cos (t-\pi) \quad\) J. \(y=-\sin t\)

Step-by-Step Solution

Answer

None of the options F, G, H, or J have the same graph as \(y=-\cos t\).

Step 1: Observe the given function

The function given is \(y = -\cos t\). This equation represents the cosine function reflected over the x-axis. So, we need to look for a function that has the same behavior.

Step 2: Analyze Option F

The first option is \(y = \cos (-t)\). This equation represents a reflection of the cosine function over the y-axis. It is not the same as reflecting over the x-axis, so it is not the correct answer.

Step 3: Analyze Option G

The second option is \(y = \sin (t - \pi)\). This equation represents a shift of the sine function to the right by \(\pi\) radians. The sine function and the cosine function have the same waveform but are out of phase by \(\pi/2\) radians, and shifting the sine function does not produce a reflection over the x-axis. So, it is also not the right answer.

Step 4: Analyze Option H

The third option is \(y = \cos (t - \pi)\). This equation is a shift of the cosine function to the right by \(\pi\) radians. It's not the same as reflecting over the x-axis and thus is not the right answer.

Step 5: Analyze Option J

The last option is \(y = -\sin t\). This equation shows a sine function reflected over the x-axis. The negative sine function is equivalent to the cosine function with a phase shift of \(\pi/2\), so it also does not produce the same graph as \(y=-\cos t\).

Key Concepts

Graph Transformation

Graph transformation refers to various modifications we can apply to the graphs of functions to obtain new, functionally related graphs. Understanding these transformations helps us easily manipulate and predict how graphs will behave under certain changes.

Common types of graph transformations include:

• Translation: Shifting the graph horizontally or vertically.
• Reflection: Flipping the graph over a particular axis. For example, reflecting a function over the x-axis changes the signs of all the y-values, resulting in an inverted graph.
• Stretching/Compressing: Altering the graph's scale vertically or horizontally, making it taller, shorter, wider, or narrower.

In the given exercise, the function \( y = -\cos t \)is a reflection of the standard cosine graph over the x-axis. This means every point on the standard cosine function has been inverted (negated) along the y-axis. As we explore the other options, it becomes clear that none of these transformations result in an identical graph to \( y = -\cos t \). Understanding these transformations enables us to predict and analyze how the graph of a function will change.

Cosine Function

The cosine function, denoted as \( \cos(t) \), is a periodic function, which means it repeats its values at regular intervals. Specifically, its period is \( 2\pi \), meaning the curve repeats every \( 2\pi \) radians.

• The graph of the cosine function starts at a maximum point (1 at \( t = 0 \)), decreases to 0, decreases further to a minimum point (-1 at \( t = \pi \)), then increases back to 0, and finally returns to the maximum point.
• The cosine function is even, which means \( \cos(-t) = \cos(t) \). This property leads to symmetry about the y-axis.

When reflecting the cosine function over the x-axis, as in \( y = -\cos t \), the maximum and minimum points switch. This means where the usual cosine function outputs a positive, the reflected function outputs a negative, effectively inverting the graph's waveform. In this particular problem, the reflection transformation changes the behavior and appearance of the function's graph but does not alter its periodic nature. The options provided in the exercise explore variations on the cosine graph through shifts and reflections, but understanding the behavior of the standard cosine function is key to identifying correct transformations.

Sine Function

Similar to the cosine function, the sine function, represented as \( y = \sin(t) \), is also periodic, with a period of \( 2\pi \). This periodic nature means it also repeats itself every \( 2\pi \) radians. However, the sine function starts at 0, reaches a maximum at \( \frac{\pi}{2} \), returns to 0 at \( \pi \), decreases to a minimum at \( \frac{3\pi}{2} \), and finally cycles back to 0 at \( 2\pi \).

Some important properties of the sine function include:

• The sine function is odd, meaning \( \sin(-t) = -\sin(t) \), which results in symmetry across the origin.
• The waveform of the sine function can be transformed similarly to the cosine function through reflections and translations.

For the exercise, the option \( y = -\sin t \) represents a sine function reflected over the x-axis. This behavior mirrors that of the \( y = -\cos t \) function in terms of flipping the waveform. While the sine and cosine functions are closely related in shape, they are offset by \( \frac{\pi}{2} \) radians. Thus, despite a reflection in option J, it is still necessary to understand this phase difference to see why it doesn't represent the \( y = -\cos t \) graph accurately.