Problem 10 - Algebra and Trigonometry

Question

Graph the function. $$ g(x)=-\frac{2}{3} \cos x $$

Step-by-Step Solution

Answer

Graph the wave with amplitude \(\frac{2}{3}\), period \(2\pi\), reflected across the x-axis.

Step 1: Identify the Function Type

The given function is a trigonometric function since it involves the cosine function, specifically: \( g(x) = -\frac{2}{3} \cos x \). This means we will graph a cosine wave which is scaled and reflected.

Step 2: Determine the Amplitude

The amplitude of \( g(x) \) is given by the absolute value of the coefficient of the cosine function, which is \( \left| -\frac{2}{3} \right| = \frac{2}{3} \). This tells us the maximum height above and below the midline (x-axis) of the graph.

Step 3: Identify the Period

For the cosine function \( \cos x \), the period is \( 2\pi \). Since there is no horizontal stretching or compression, the period of \( g(x) = -\frac{2}{3} \cos x \) remains \( 2\pi \).

Step 4: Determine the Phase Shift

There is no horizontal shift in this function as there is no horizontal translation term added or subtracted inside the cosine function.

Step 5: Determine the Vertical Shift

There is no vertical shift in the function since nothing is added or subtracted outside the cosine term. Thus, its midline remains at \( y = 0 \).

Step 6: Reflect and Graph the Function

Since the function is \( g(x) = -\frac{2}{3} \cos x \), the graph will be a reflection of the cosine function across the x-axis. Start by plotting critical points: at \( x = 0 \), \( g(x) = -\frac{2}{3} \); at \( x = \pi \), \( g(x) = \frac{2}{3} \); at \( x = 2\pi \), \( g(x) = -\frac{2}{3} \). Draw the wave connecting these points smoothly, between each critical point, crossing the x-axis halfway.

Step 7: Verify the Characteristics

Ensure that the amplitude is \( \frac{2}{3} \), the period is \( 2\pi \), the peak points reflect the negative cosine function, and the wave repeats every \( 2\pi \). These verify that the graph accurately represents \( g(x) = -\frac{2}{3} \cos x \).

Key Concepts

Understanding Amplitude

In trigonometric functions, amplitude refers to the height of the wave from its midline. For many functions like cosine and sine, it's always positive and indicates how much the function oscillates.
This amplitude is determined by the coefficient of the trigonometric function. In the case of the function \( g(x) = -\frac{2}{3} \cos x \), the amplitude is the absolute value of \( -\frac{2}{3} \), which happens to be \( \frac{2}{3} \).

• The amplitude describes the maximum displacement from the midline.
• A larger amplitude means taller peaks and deeper troughs.
• Graphs of cosine and sine functions have their midlines at \( y = 0 \).

Thus, for our function, the graph will vary \( \frac{2}{3} \) units above and below the x-axis, which is the midline for this cosine wave.

Period of Trigonometric Functions

The period of a trigonometric function defines how long it takes for the wave to complete one full cycle before repeating. For the basic cosine function \( \cos x \), this period is \( 2\pi \).

• This means it takes \( 2\pi \) units along the x-axis for the wave to start repeating itself.
• If the function has a different form, such as \( \cos(bx) \), the period changes to \( \frac{2\pi}{b} \).

In our function \( g(x) = -\frac{2}{3} \cos x \), there is no horizontal stretch or compression, so the period remains \( 2\pi \). This means as you graph from \( x = 0 \) to \( x = 2\pi \), you will complete exactly one full wave of the cosine function.

Reflection of Cosine Function

Reflection in the graph of trigonometric functions occurs when you multiply the function by a negative sign. For the function \( g(x) = -\frac{2}{3} \cos x \), this multiplication causes the graph to flip vertically over the x-axis.

• Normally, a positive cosine graph starts at its maximum value at \( x = 0 \).
• When reflected, the graph starts at its minimum value at the same point.

For our function, at \( x = 0 \), instead of starting at the positive amplitude \( \frac{2}{3} \), it begins at \( -\frac{2}{3} \).

• As you move towards \( x = \pi \), instead of hitting a minimum, the reflected cosine hits a maximum.
• Hence, the function travels back to its starting point by \( x = 2\pi \).

This reflection results in a wave that mirrors the usual cosine function from top to bottom, giving the unique form of our function's graph.