Problem 47 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Graph the function. \(g(x)=-4 \cos \left(x+\frac{\pi}{4}\right)-1\)

Step-by-Step Solution

Answer

The graph of \( g(x)=-4 \cos \left(x+\frac{\pi}{4}\right)-1 \) has an amplitude of 4, period of \( 2\pi \), a left shift of \( \frac{\pi}{4} \), and moves down by a length of one.

Step 1: Identify Amplitude and Period

The amplitude of a cosine function is given by the absolute value of the coefficient of the function. In this case, the amplitude is 4. The period of a cosine function is calculated as \( 2\pi \) divided by the absolute value of the coefficient of \( x \), which is 1 in this case. So, the period is \( 2\pi \).

Step 2: Identify Phase Shift and Vertical Shift

The phase shift of the function is given by the value subtracted or added inside the cosine function. In this case, the phase shift is \( -\frac{\pi}{4} \), which means it's shifted to the left by \( \frac{\pi}{4} \) units. The vertical shift is given by the value added or subtracted to the function, and in this case is -1, which means it will move down by a length of one.

Step 3: Sketch the Graph

With the amplitude, period, phase shift, and vertical shift identified, the graph can be sketched. The graph will have peaks at -4 and -1, and will repeat every \( 2\pi \) units. Applying the phase shift means the graph will start on the left by \( \frac{\pi}{4} \) units. Finally, the graph is shifted down by one unit.

Step 4: Check

Plot a few points to check if the graph has been sketched correctly. For example, when \( x = 0 \), the value of the function will be \( -4 \cos(\frac{\pi}{4}) - 1 \). Plot a few more points and make sure that they match with the sketched graph.

Key Concepts

Amplitude

The amplitude of a function tells us how far the peaks and troughs of the wave extend from its central axis. For the cosine function, the amplitude is determined by the coefficient in front of the cosine term. In the given function, \(g(x) = -4 \cos \left(x + \frac{\pi}{4}\right) - 1\), the coefficient is \(-4\). Since amplitude is always a positive value (representing distance), we take the absolute value, which is \(4\). This means the wave will rise and fall a maximum of 4 units from its center.
Amplitude helps us understand the scale of fluctuations in the cosine function. Thus, in our function, expect the wave to reach as high as 4 units above and below the midline.

Period

The period of a function indicates how long it takes for the function to complete one full cycle before repeating itself. For a standard cosine function, the period is \(2\pi\).
In our function, the coefficient of \(x\) inside the cosine is 1, meaning the function has not been stretched or compressed horizontally. Therefore, its period remains \(2\pi\).

• This means that every \(2\pi\) units along the \(x\)-axis, the wave repeats its pattern.

Understanding the period is crucial as it helps predict where peaks and valleys will occur on the graph.

Cosine Function

The cosine function is a fundamental trigonometric function, often represented as \(\cos(x)\). It is periodic with a repeating wave form.
Cosine functions characterize many natural phenomena, such as sound waves and tides. In our example, the function is \(-4 \cos \left(x + \frac{\pi}{4}\right) - 1\).

• The graph of a cosine function oscillates symmetrically around a central axis, making it easier to apply transformations like amplitude and period adjustments.

In the given function, the cosine wave is negative, meaning the wave is inverted, and the peaks become troughs at their respective positions.

Phase Shift

A phase shift occurs when a function is horizontally moved left or right along the \(x\)-axis. The value inside the cosine function which adds or subtracts shifts its starting point.
In the function \(g(x)=-4 \cos \left(x + \frac{\pi}{4}\right) - 1\), the expression \(x + \frac{\pi}{4}\) indicates a phase shift.

• The phase shift value is \(-\frac{\pi}{4}\), meaning the cosine wave starts \(\frac{\pi}{4}\) units to the left of the usual starting point \(x = 0\).

Recognizing phase shifts helps position the wave's cycle accurately.

Vertical Shift

Vertical shifts adjust the entire graph up or down along the \(y\)-axis. In our function, the term \(-1\) signifies a shift. This negative sign indicates moving the entire graph downward by one unit.
This transforms the midline or central axis of the wave. In a typical cosine function, the midline is \(y = 0\). With a vertical shift of \(-1\), the midline now becomes \(y = -1\).

• This adjustment changes where peak and trough points appear relative to the \(y\)-axis.

Understanding the vertical shift is vital for accurately sketching and interpreting the overall graph's position.