Problem 84 - Algebra 2

Question

Find the amplitude and period of each function. Describe any phase shift and vertical shift in the graph. $$ y=-\cos (x+4)-7 $$

Step-by-Step Solution

Answer

The amplitude of the function is 1, the period is \( 2\pi \), there is a phase shift of 4 units to the left and a vertical downward shift of 7 units.

Step 1: Identify the Amplitude

The amplitude of a function is the absolute value of the coefficient of the trigonometric function. In our expression, the coefficient is -1, thus, the amplitude of the function is \( |-1| = 1 \).

Step 2: Identify the Period

The period of the cos function without any modification is \( 2\pi \). However, no transformation changes the period in this function, so the period remains \( 2\pi \).

Step 3: Identify the Phase Shift

Phase shift is determined by the term inside the cosine function. The shift occurs along the x-axis. We see \( x+4 \) suggesting a phase shift of 4 units to the left, which means our graph starts from the point \( x=-4 \).

Step 4: Identify the Vertical Shift

Vertical shift refers to the term outside the cosine function which shifts the graph upwards or downwards. Here we have -7 outside the cosine function, which indicates a downward shift of 7 units. Hence, the entire graph will be pulled down by 7 units.

Key Concepts

Understanding Amplitude

In trigonometric functions, amplitude represents the maximum value a function can achieve from its central or equilibrium position. It's essentially the height of a wave from its midline to its peak or trough. In simpler terms, think of amplitude as how "tall" a wave is. For the cosine function \( y = -\cos(x+4)-7 \), the coefficient in front of the cosine term determines this amplitude. Here, it's \(-1\). To find the amplitude, we take the absolute value of this coefficient. Thus, the amplitude is \(|-1| = 1\).
This tells us that no matter how the function shifts or transforms, it always moves 1 unit away from its center. Even negative coefficients only reflect the wave, not change its height.

Decoding the Period

The period of a trigonometric function is the length it takes for the function to complete one full cycle. For basic cosine and sine functions, this period is \(2\pi\). In the function \( y = -\cos(x+4)-7 \), there is no additional factor multiplying \(x\) inside the cosine, meaning no horizontal stretch or compression affects the period.

• The period remains at \(2\pi\).

This implies that every \(2\pi\) units along the x-axis, the function repeats its pattern. The regularity of this cycle is crucial for predicting the wave's behavior over time, making it fundamental in fields like physics and engineering.

Navigating Phase Shift

Phase shift in trigonometry involves the horizontal movement of a wave along the x-axis. This determines where the wave starts its cycle. For \( y = -\cos(x+4)-7 \), the expression inside the cosine function \((x+4)\) suggests a phase shift. Since it reads \(x+4\), we equate \(x+4 = 0\) to determine the starting point of the wave, yielding \(x = -4\).
Thus, the graph is shifted 4 units to the left.

• Phase shifts are common when comparing different wave functions or signals, helping to understand how two different waves may be in or out of sync with one another.

Grasping Vertical Shift

Vertical shift concerns the up or down movement of a wave on the graph. It adjusts the baseline or middle line of the wave. In the equation \( y = -\cos(x+4)-7 \), the \(-7\) indicates this vertical shift. It translates the entire function 7 units downward.
A vertical shift modifies the average value around which the wave oscillates. In practical terms, think of it as moving the entire wave up or down a graphing grid without altering its shape. This concept is crucial in matching a trigonometric function to real-world processes where the baseline may vary due to external factors, like tides influenced by seasonal changes.