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Writing to Learn: In a certain video game a cursor bounces back and forth horizontally across the screen at a constant rate. Its distance d from the center of the screen varies with time t and hence can be described as a function of t. Explain why this horizontal distance d from the center of the screen does not vary according to an equation d= a sin bt, where t represents seconds. You may find it helpful to include a graph in your explanation

To answer this question, we need to understand the properties of the function d = a sin(bt) and how it relates to the given scenario of a cursor bouncing back and forth horizontally across a screen.

Let's first have a look at the function d = a sin(bt). This is a sine function, where a determines the amplitude (or maximum displacement) of the oscillation, and b determines the frequency or rate of oscillation. The function is periodic, meaning it repeats itself after a certain interval. In this case, t represents time in seconds.

Now, let's consider the scenario of a cursor bouncing back and forth horizontally across a screen. When the cursor moves towards the right (positive direction) from the center, it covers a certain distance and then changes its direction to the left (negative direction). This process repeats over time, forming a back-and-forth motion.

If we graph the function d = a sin(bt) using time (t) on the x-axis and distance (d) on the y-axis, we will observe a smooth wave-like pattern. The sine function allows for continuous changes in distance as time progresses. However, in the given scenario, the cursor's motion is not continuous. It moves at a constant rate from one end of the screen to the other and then abruptly changes direction and repeats the process.

Graphically, this scenario would be represented by a series of line segments connecting the cursor's positions as it moves horizontally across the screen. This is in contrast to the smooth, wavelike oscillation shown by the sine function.

Therefore, the equation d = a sin(bt) is not an appropriate representation for the horizontal distance of the cursor from the center of the screen in this scenario. The cursor's motion can be better described using a piecewise function that represents the constant back-and-forth motion across the screen.

In summary, the given scenario of a cursor bouncing back and forth horizontally across a screen does not follow the pattern of a sine function. Therefore, the equation d = a sin(bt) does not accurately describe the variation of the distance from the center of the screen.