Problem 66 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\). \(y=\cos (x+\pi)-3\)

Step-by-Step Solution

Answer

The graph of \(y=\cos (x+\pi)-3\) between 0 and \(2\pi\) is a cosine curve shifted left by \(\pi\) and downward by 3 units.

Step 1: Understand the parent function

The main function we are dealing with is the cosine function, denoted by \(y=\cos(x)\). Cosine starts at (0,1), oscillates between 1 and -1 and has a period of \(2\pi\). It makes a complete cycle between 0 and \(2\pi\).

Step 2: Apply horizontal shift

The function \(y=\cos (x+\pi)\) represents a horizontal shift to the left by \(\pi\) units of the cosine graph. When \(x=0\), it results in \(y=\cos (\pi)\), which equals -1.

Step 3: Apply vertical translation

The graph \(y=\cos (x+\pi)-3\) indicates a vertical shift of the cosine graph downwards by 3 units. This means every point of the original cosine graph will move 3 units downward in the resultant graph.

Step 4: Draw the final graph

Now, draw the graph of the function \(y=\cos (x+\pi)-3\) on the interval [0, \(2\pi\)]. Mark the important points and draw the curve going through these points, making sure the curve shows the downward shift and the cycle of the cosine function.

Key Concepts

Understanding the Cosine Function

The cosine function, represented as \(y = \cos(x)\), is a fundamental trigonometric function. It is periodic with a period of \(2\pi\), meaning it repeats its values in regular intervals of \(2\pi\). The graph of the cosine function has a distinctive wave-like shape, starting from the high point 1 at \(x = 0\). As you move along the x-axis within one period, it dips to -1 and returns to 1, completing one full cycle.
Cosine achieves its maximum value of 1 when \(x=0\), then proceeds toward its minimum value of -1 at \(x = \pi\), and returns back to 1 at \(x = 2\pi\). This consistent oscillation between 1 and -1 forms the classic cosine wave, which is symmetric about the y-axis.
Key properties of the cosine function include:

• Amplitude: 1 (the peak height from the centerline)
• Period: \(2\pi\) (the length of one complete cycle)
• Midline: \(y = 0\) (the horizontal line exactly between the peaks and troughs)

Understanding these characteristics helps in predicting and sketching the cosine graph in varied translations.

Horizontal Translation of the Cosine Function

Horizontal translation involves shifting the graph of a function left or right across the x-axis. For the cosine function, a horizontal shift is achieved by adding or subtracting a constant inside the function's argument. In the expression \(y = \cos(x + \pi)\), the \(+\pi\) inside the parentheses indicates a shift to the left by \(\pi\) units.
When a constant is added to \(x\) (e.g., \(x + c\)), the shift moves to the left, whereas subtracting (e.g., \(x - c\)) moves the graph to the right. This horizontal movement does not change the graph's shape, amplitude, or period, just the starting point of its cycle.
In our case, starting at \(x = 0\), the original cosine function normally equals 1, but with \(y = \cos(x + \pi)\), it shifts to equate to its value at \(\cos(\pi)\), which equals -1. This shift effectively moves each point of the cosine curve left, maintaining the periodic properties but altering where the cycle begins.

Vertical Translation of the Cosine Function

A vertical translation modifies how high or low a graph is positioned relative to the x-axis. It moves every point on the graph up or down by a specific number of units. In the function \(y = \cos(x + \pi) - 3\), the \(-3\) indicates a downward shift by 3 units across the whole graph.
Vertical translation is implemented by adding or subtracting a constant from the entire cosine function result, not inside the argument. In our specific function, you decrease each point by 3, moving the midline from \(y = 0\) to \(y = -3\) and adjusting its amplitude accordingly.
This adjustment means:

• The maximum value shifts from 1 to -2, calculated as \(1 - 3\).
• The minimum value shifts from -1 to -4, calculated as \(-1 - 3\).

Such vertical changes affect the range of the cosine function's output values, extending the graph's reach on the y-axis but leaving the period and horizontal alignment untouched.