Problem 20 - Algebra 2

Question

Describe any phase shift and vertical shift in the graph. $$ y=\cos (x-3)+2 $$

Step-by-Step Solution

Answer

The graph of the function \(y=\cos (x-3)+2\) has a phase shift of 3 units to the right and a vertical shift of 2 units upwards.

Step 1: Identify the Phase Shift

The phase shift of a function is defined by the value of 'x' in the formula. In the given function \(y=\cos (x-3)+2\), the phase (or horizontal) shift is given by '3'. However, because it is '(x-3)', the direction of the shift is to the right. So, the graph is shifted 3 units to the right.

Step 2: Identify the Vertical Shift

The vertical shift of a function is determined by the value added to or subtracted from the function. In this case, '+2' is added to \(cos (x-3)\), which means that the graph is shifted upwards by 2 units.

Key Concepts

Phase Shift

The term 'phase shift' refers to the horizontal movement of a wave or function along the x-axis. This tells us how far the graph of the function is shifted from its usual starting point. For a cosine function like \(y=\cos (x-3)+2\), identifying the phase shift is straightforward.

When the function is set up as \(y = \cos(x - b)\), the 'b' gives the phase shift. Here, it's a shift to the right if \(b\) is positive and to the left if \(b\) is negative. In our example, \(b = 3\), indicating that the graph moves 3 units to the right.

When interpreting phase shifts:

• If it's in the form \((x - b)\), move right.
• If it's in the form \((x + b)\), move left.

Knowing this can help you easily adjust the position of any trigonometric function on a graph.

Vertical Shift

The vertical shift is all about moving the entire graph up or down. This concept changes the function's position along the y-axis. When looking at a function like \(y = \cos(x - 3) + 2\), the vertical shift is determined by the number added or subtracted at the end.

In this case, we add '+2', meaning the whole graph of the function shifts upward by 2 units. Adding moves the graph upwards, while subtracting would lower it.

Here's what you should remember about vertical shifts:

• Adding a positive number shifts the graph up.
• Subtracting a number shifts it down.

This makes it clear how the peaks and valleys of the cosine function are elevated by the shifts in the equation.

Cosine Function

The cosine function is a fundamental trigonometric function that creates a wave-like pattern on a graph. The standard form is expressed as \(y = \cos x\). It regularly cycles, creating peaks and valleys, mimicking the motion seen in waves or vibrations.

The function \(y = \cos(x - 3) + 2\) shows us how changes in the formula modify the graph's appearance:

• The cosine wave starts at 1, moves to -1, and back, repeating this cycle.
• Horizontal stretches or squeezes can occur if multiplied by a constant, like \(c\cos(x)\).

The adjustments like phase and vertical shifts distort this standard form. The phase shift moves it horizontally, while the vertical shift elevates or lowers it. Together, these transformations provide flexibility in modeling cycles in real-world phenomena, such as sound waves or tides, with the cosine function.