Problem 60 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Consider the functions \(y=\sin (-x)\) and \(y=\cos (-x)\) a. Construct a table of values for each equation using the quadrantal angles in the interval \(-2 \pi \leq x \leq 2 \pi\) b. Graph each function. c. Describe the transformations of the graphs of the parent functions.

Step-by-Step Solution

Answer

The table of values shows the y-values of the functions \(y = \sin(-x)\) and \(y = \cos(-x)\) for certain x-values in the range \(-2\pi, 2\pi\). The key points (peaks, troughs, and zeros) of \(\sin\) and \(\cos\) functions occur at these x-values. Graphing these functions shows a y-axis reflection of the parent functions \(y = \sin(x)\) and \(y = \cos(x)\). Furthermore, the waveform shape of the parent functions has been retained.

Step 1: Construct a Table of Values

For the range \(-2 \pi \leq x \leq 2 \pi\), the selected values for x are: \(-2\pi, -3\pi/2, -\pi, -\pi/2, 0, \pi/2, \pi, 3\pi/2, 2\pi\). Substitute these values into the functions \(y = \sin(-x)\) and \(y = \cos(-x)\) to get the corresponding y-values.

Step 2: Graph the Functions

Based on the table of values generated in the Step 1, plot the points on a Cartesian coordinate system. Note that the x-axis represents the angle in radians and the y-axis represents the values of the sine and cosine functions.

Step 3: Describe the Transformations

Look at the original parent functions \(y = \sin(x)\) and \(y = \cos(x)\) and compare them with the graphed functions to identify any shifts, stretches, or reflections. The transformation is a reflection in the y-axis because of the negative sign in front of x in both functions. The graph of \(y = \sin(-x)\) and \(y = \cos(-x)\) is a reflection of the graphs of \(y = \sin(x)\) and \(y = \cos(x)\), respectively, about the y-axis.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can initially seem complex, but by breaking it down into easy steps, you'll find it's manageable and even fun! When graphing functions like \( y = \sin(-x) \) and \( y = \cos(-x) \), the first thing to do is create a table of values. This table will help you understand how each function behaves within a specific range.

• First, choose the values for \( x \). For example, \(-2\pi\) to \(2\pi\) can be a good starting point as it covers one full cycle of these periodic functions.
• Calculate \( y \) by substituting these \( x \) values into your given function. You should find that they repeat their values in a systematic way, unveiling the periodic nature of trigonometric functions.
• Plot these points on a graph, using the x-axis for angles (usually in radians) and the y-axis for function values.

This process allows you to visualize the shape and pattern of the sine and cosine waves, revealing core characteristics of trigonometric functions such as amplitude, period, and frequency.
Understanding how to graph these functions is an essential skill that helps you solve more complex mathematical problems.

Sine and Cosine Transformations

Transformations change how the basic graphs of sine and cosine look. You may remember from earlier studies that the graphs of \( y = \sin(x) \) and \( y = \cos(x) \) are smooth periodic waves, undulating along the x-axis. However, with transformations, we can modify these parent functions to achieve various results.

• Changing the coefficient of \( x \) inside the sine or cosine function can stretch or compress the waveform horizontally.
• Altering values outside the function, such as multiplying the function by two, impacts the amplitude, making the waves taller.
• Adding or subtracting from the entire function shifts the graph up or down, impacting its vertical position.

These transformations are crucial as they affect how the functions map real-world periodic phenomena, such as sound waves or tides. A visual approach by graphing each function and then comparing their shapes helps students grasp these transformations accurately.
Being aware of how transformations affect the sine and cosine functions offers expanded tools for solving equations and interpreting results in both academic and real-world scenarios.

Reflections in the y-axis

One of the transformations for trigonometric functions involves reflecting them. When a function is reflected, its graph is flipped over a certain line, like the y-axis. This is exactly what happens with \( y = \sin(-x) \) and \( y = \cos(-x) \).

• A reflection across the y-axis occurs when every point of the graph of \( y = \sin(x) \) or \( y = \cos(x) \) that was on the right of the y-axis now appears directly across it on the left, and vice versa.
• To identify this reflection when graphed, notice the symmetry; the shape is unchanged except for this directional turnover.
• In the case of trigonometric functions, a reflection typically doesn't alter the amplitude or period, only the phase (or horizontal shift).

Understanding reflections are vital in recognizing and predicting how functions behave under certain transformations.
This concept showcases the versatility of mathematical functions like sine and cosine, allowing us to model multiple symmetrical scenarios as seen in nature and engineering.