Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=\cos (-2 x) $$

Short answer

Domain: \((-\infty, \infty)\). Range: \([-1, 1]\).

Step-by-step solution

Understand the Basic Cosine Function

The basic function to consider is the cosine function, given by: \[ y = \cos(x) \]. This function has a period of \( 2\pi \), a range of \([-1, 1]\), and key points at \( x = 0, \pi/2, \pi, 3\pi/2, 2\pi \).

Transformation by Horizontal Scaling

The given function is \( y = \cos(-2x) \). The factor \(-2\) affects the x-coordinates of the key points. It's a horizontal compression by a factor of 2 (because the period changes from \( 2\pi \) to \( \pi \) due to the factor 2), and the negative sign indicates a reflection over the y-axis.

Identify Key Points of Transformed Function

Since the period changes to \(\pi\), recalculate the key points: For \(x = 0\), \( y = \cos 0 = 1\)For \(x = \pi/4\), \( y = \cos(-2 \pi/4) = \cos(-\pi/2) = 0\)For \( x = \pi/2\), \( y = \cos(-2 \pi/2) = \cos(-\pi) = -1 \)For \( x = 3\pi/4\), \( y = \cos(-2 \cdot 3\pi/4) = \cos(-3\pi/2) = 0 \)For \( x = \pi \), \( y = \cos(-2\pi) = 1 \).

Plot the Points and Graph the Function

Plot these key points on the graph: (0, 1), (\pi/4, 0), (\pi/2, -1), (3\pi/4, 0), and (\pi,1). Connect these points smoothly to show the graph of one cycle. Since we want to show two cycles, repeat the pattern for the next cycle by adding \(\pi\) to each x-coordinate.

Determine the Domain and Range

The domain of \( y = \cos(-2x) \) is all real numbers, \( (-\infty, \infty) \), as cosine is defined for all x-values. The range remains \([-1, 1]\) since the amplitude does not change with transformations.

Key concepts

Horizontal Scaling

Horizontal scaling is a transformation applied to the x-coordinates of the points on a graph. If we consider the function \(y = \cos(x)\), any transformation that involves a multiplicative factor applied to x is a horizontal scaling.
For the function \(y = \cos(-2x)\), the factor of \-2\ affects how the graph is stretched or compressed horizontally.
Specifically, the \-2\ signifies a horizontal compression by a factor of 2 and reflection over the y-axis.
Horizontal scaling modifies the distance between key points on the graph without changing the amplitude.

• For a function \(y = \cos(bx)\), if \|b| > 1\, the graph is compressed horizontally.
• If \|b| < 1\, the graph is stretched horizontally.
• The negative sign indicates reflection over the y-axis.

In our specific example, this transforms the period of the cosine function.

Function Transformations

Function transformations involve modifying a function in a systematic way, often to better understand its behavior or fit a specific need.
There are different types of transformations such as translation, reflection, and scaling.

• **Translation** involves shifting the graph horizontally or vertically without changing its shape.
• **Reflection** involves flipping the graph over a specified line, like the x-axis or y-axis.
• **Scaling**, as mentioned before, stretches or compresses the graph either horizontally or vertically.

In our problem, \(y = \cos(-2x)\) is a transformation by reflection and horizontal scaling. The negative sign inside the cosine turns the graph upside down, reflecting over the y-axis. Meanwhile, the factor of 2 inside the cosine compresses the graph horizontally. Remembering these kinds of transformations helps in graphing functions and understanding their properties.

Period of Trigonometric Functions

The period of a trigonometric function is the interval over which the function repeats itself. For the basic cosine function, \(y = \cos(x)\), the period is \(2\pi\).
This means every \(2\pi\) units along the x-axis, the function's values repeat.
When we transform the cosine function, its period changes accordingly.
Given the function \(y = \cos(-2x)\), we need to find how the period is affected by the constant \-2\.
The period of \(y = \cos(bx)\) is given by \[ \frac{2\pi}{|b|} \].
For our function, \frac{2\pi}{|2|} = \frac{2\pi}{2} = \pi .\ Therefore, the period of \(y = \cos(-2x)\) is \pi\.
The cosine function now completes one full cycle within a \pi\ interval. Understanding how to determine the period of trigonometric functions is crucial for graphing and analyzing their behavior across different transformations.