Question

\(y=\frac{1}{2} \cos 0.75 x-8\)

Short answer

The graph of the function \(y=\frac{1}{2} \cos 0.75x -8\) has an amplitude of 0.5, a period of approximately 8.38, no phase shift, and a vertical shift of -8.

Step-by-step solution

Determine the Amplitude

The amplitude of a cosine function is determined by the coefficient in front of the cosine term. In this function, the coefficient is 0.5, so the amplitude is 0.5.

Calculate the Period

The period of a cosine function is determined by dividing \(2\pi\) by the absolute value of the coefficient inside the cosine function. Here that coefficient is 0.75, so the period is \( \frac{2\pi}{|0.75|} \approx 8.38 \).

Find the Phase Shift

Since there is no addition or subtraction inside the cosine function, there is no phase shift. The graph starts at its peak when x is equal to 0.

Find the Vertical Shift

The vertical shift is determined by the constant at the end of the function. Here, the constant is -8, so the entire graph should be shifted 8 units downwards.

Sketch the Graph

Now with all the values identified, draw a graph with a peak at -7.5, a trough at -8.5, a period of approximately 8.38, and no phase shift.

Key concepts

Amplitude

Amplitude is a key feature of trigonometric functions like cosine, defining how far the graph stretches vertically from its central axis. It represents the maximum value that the function can achieve from its midline.
In mathematical terms, for a function of the form \[ y = a \, \cos(bx + c) + d \], the amplitude is given by the absolute value of \(a\).
This means our function, \( y = \frac{1}{2} \, \cos(0.75x) - 8 \), has an amplitude of \(0.5\). This is because the term in front of the cosine function is \(\frac{1}{2}\).

• A larger amplitude implies taller peaks and deeper troughs.
• A smaller amplitude indicates a flatter wave.

The amplitude never affects the horizontal stretching or period of the function.
Understanding amplitude helps in predicting the overall extent of variation in the wave's height.

Period of Cosine Function

The period of a cosine function informs us of how long it takes for the graph to complete one full cycle and repeat itself. This period is determined by the coefficient that multiplies the variable inside the cosine function.
For the general cosine function \( y = a \, \cos(bx + c) + d \), the period is calculated using the formula\[ \frac{2\pi}{|b|} \].
Applying this to our function \( y = \frac{1}{2} \, \cos(0.75x) - 8 \), we take the coefficient inside the cosine, which is \(0.75\). This gives us a period of \( \frac{2\pi}{0.75} \), approximately \(8.38\).

• A higher coefficient means a shorter period, leading to more cycles over the same horizontal space.
• A lower coefficient results in a longer period, indicating fewer cycles.

The period is crucial for understanding how often the wave repeats along the horizontal axis.

Vertical Shift

In trigonometric functions, the vertical shift refers to how far up or down the entire graph is moved on the coordinate plane. This shift is determined by the constant term added or subtracted from the main function.
In the function \( y = a \, \cos(bx + c) + d \), the vertical shift is represented by \(d\).
For the given function \( y = \frac{1}{2} \, \cos(0.75x) - 8 \), the graph is shifted vertically downward by 8 units due to the \(-8\) at the end.

• A positive shift moves the graph upwards.
• A negative shift moves it downwards.

The midline of the graph, which would normally be at \(y=0\), is now at \(y=-8\).
Vertical shifts do not affect the amplitude or period, but they do dictate where the midline of the wave is located on the graph.

Graph Sketching

Sketching trigonometric graphs involves combining what we know about amplitude, period, and vertical shift.
For our cosine function \( y = \frac{1}{2} \, \cos(0.75x) - 8 \), these components tell us:

• The amplitude of \(0.5\) indicates peaks will reach half a unit above the midline and troughs half a unit below it.
• The period of \(8.38\) informs the length of one complete wave cycle on the x-axis.
• The vertical shift of \(-8\) positions the midline at \(y = -8\).

To sketch the graph:

• Start the cycle at its peak at \(x=0\), since there is no phase shift.
• Draw one wave cycle over the interval from \(x=0\) to \(x=8.38\).
• Position the peaks at \(-7.5\) and the troughs at \(-8.5\), vertically oscillating around the shifted midline at \(y=-8\).

This visualization helps in demystifying how combinations of different terms in a trigonometric function come together to form complex wave patterns.