Problem 12 - Algebra 2

Question

Graph each translation of \(y=\cos x\) in the interval from 0 to 2\(\pi\) $$ y=\cos x+3 $$

Step-by-Step Solution

Answer

The graph of \(y=\cos x + 3\) is a vertically translated version of the graph of \(y=\cos x\), shifted up by three units. It oscillates in the same manner as the original cosine graph and within the interval from 0 to \(2\pi\), just three units higher.

Step 1: Draw the Original Function

Establish a graph with both an x and y axis. On the x-axis, mark out the interval from 0 to \(2\pi\). Draw the graph of \(y = \cos x\) on this interval, remembering that the graph of the cosine function starts at its maximum value at \(x = 0\), descends to its minimum value halfway through the interval, and rises back to its maximum at \(x = 2\pi\)

Step 2: Understand the Translation

In the translation \(y = \cos x + 3\), the \(+ 3\) adds onto each y-coordinate of \(y = \cos x\). This means that at every point along the x-axis, the translated graph will be three units higher than the original. Hence, the effect of adding 3 to the cosine function is to shift the entire graph upward by three units.

Step 3: Draw the Translated Graph

Using the same x-axis interval from 0 to \(2\pi\), draw the graph of \(y = \cos x + 3\). The minimum point of this graph will be 3 units above the minimum point of the original cosine graph, and the maximum point will be 3 units above the maximum point of the original cosine graph. The graph will still oscillate in the same manner as the original cosine graph; it will just be shifted three units higher.

Key Concepts

Cosine Function

The cosine function, denoted as \(y = \cos x\), is a fundamental trigonometric function that describes a smooth, wavelike pattern. It is periodic, with a period of \(2\pi\), meaning it repeats its shape every \(2\pi\) units along the x-axis. This function starts at its maximum value of 1 when \(x = 0\), decreases to its minimum value of -1 at \(x = \pi\), and returns to 1 at \(x = 2\pi\).

Key features of the cosine function include:

• The amplitude, which is the height from the centerline to the peak, usually 1 in the basic function.
• A period of \(2\pi\).
• A smooth and continuous wave, symmetrical about the y-axis.

Understanding these characteristics helps to graph cosine functions accurately and recognize their transformations.

Translations

Translations involve shifting a graph horizontally or vertically. In the context of the cosine function, a translation changes the graph's position without altering its shape. This exercise focuses on a vertical translation.

When we translate \(y = \cos x\) to \(y = \cos x + 3\), we perform a vertical shift. The \(+3\) means every point on the cosine graph moves up by 3 units on the y-axis.

The key steps for translating a graph vertically:

• Look at the function to identify the shift, here the \(+3\) indicates a vertical move.
• Adjust each y-coordinate of the original graph up or down by the translation value.

This process helps you visualize how the entire wave-like graph lifts up without changing its period or amplitude.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cosine is about understanding the function’s basic shape and how any transformations, such as translations, affect it. To accurately graph \(y = \cos x + 3\), start by sketching \(y = \cos x\) and then apply the translation.

Steps for graphing the function:

• Draw \(y = \cos x\) across the interval from 0 to \(2\pi\).
• Identify key points: maximum (1 at \(0\) and \(2\pi\)), minimum (-1 at \(\pi\)).
• Shift these points up by 3 units.
• Redraw the smooth curve through these shifted points.

By understanding these techniques, graphing shifted trigonometric functions becomes straightforward, enabling you to explore more complex alterations with confidence.