Problem 13 - Algebra 2

Question

Graph each translation of \(y=\cos x\) in the interval from 0 to 2\(\pi\) $$ y=\cos x-4 $$

Step-by-Step Solution

Answer

The graph of \(y=\cos x - 4\) is a standard cosine curve shifted down 4 units from the original cosine function \(y=\cos x\).

Step 1: Graph the original function

First, plot the original cosine function \(y=\cos x\) in the interval [0, 2\(\pi\)]. The graph of the cosine function typically starts at a maximum point, decreases to a minimum point, then returns to the maximum, repeating this pattern periodically. In the range [0, 2\(\pi\)], you should see one complete wave.

Step 2: Apply the translation

The given function to graph is \(y=\cos x - 4\), which represents a vertical translation of the basic cosine function by 4 units down. This means that every point on the original graph \(y=\cos x\) will be moved down by 4 units. So, subtract 4 from the y-coordinate of each point on the original function to derive the graph of the translated function.

Step 3: Draw the translated graph

After applying the transformation, draw the graph of \(y=\cos x - 4\). This graph should look like the original cosine function, but shifted 4 units down. The maximum point of the wave should be at \(y = -3\) (since the maximum of the original cosine function is 1 and 1 - 4 = -3), the minimum at \(y = -5\) (since the minimum of the original cosine function is -1 and -1 - 4 = -5).

Key Concepts

Graph Transformations

Graph transformations are techniques that allow us to modify the basic form of a graph. These transformations change the position, shape, or size of the graph in a systematic way. For a function like the cosine function, transformations can include translations, reflections, stretches, and compressions. Each transformation has a distinct effect on the graph.

• Translations move the graph horizontally or vertically without altering its shape.
• Reflections flip the graph over a specific axis.
• Stretches and compressions change the height and width of the graph.

Understanding transformations helps in predicting how graphs will look after changes are made. They are especially useful in trigonometry, where functions like sine and cosine have predictable wave patterns. By mastering graph transformations, you gain better control over solving problems involving function graphs.

Vertical Translation

A vertical translation shifts a graph up or down on the coordinate plane. This type of transformation affects the y-axis values of the function while leaving the x-axis values unchanged. When you vertically translate a function, you either add or subtract a constant from it. This constant value determines the direction and magnitude of the shift.

For the cosine function, a vertical translation can be represented as:

• If you have the function of the form: \( y = \,\cos x + c \), the graph is shifted upward by \( c \) units.
• For the function \( y = \,\cos x - c \), the graph is shifted downward by \( c \) units.

In the given exercise, the function is \( y = \,\cos x - 4 \). Here, the graph of the cosine function is translated down 4 units. This means each y-value on the original graph decreases by 4, effectively moving all points lower on the graph. Vertical translations are straightforward because they only require adjusting the y-coordinates.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in studying periodic phenomena. They include sine, cosine, tangent, and their reciprocals. These functions are defined based on the angles and are periodic, meaning they repeat their values in regular intervals.
The cosine function, \( y = \,\cos x \), is one of the primary trigonometric functions, and it has a distinct wave-like pattern:

• The cosine wave starts at its peak, descends to its lowest point, and returns to the peak, completing one cycle in an interval of \( 2\pi \).
• Its standard range is from -1 to 1, and it repeats every \( 2\pi \) radians.
• Key points within this interval are at \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \), where the function reaches its maximum, zero, minimum, and zero values again.

Understanding these properties helps in graphing and transforming trigonometric functions. For graph transformations, such as translations or reflections, it's crucial to know how these affect the original wave pattern. Mastery of trigonometric functions is vital, as they apply to various fields such as engineering, physics, and even economics, where periodic changes are common.