Problem 71 - Precalculus Essentials

Question

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=-2 \sin x, g(x)=\sin 2 x, h(x)=(f+g)(x)$$

Step-by-Step Solution

Answer

The functions \(f(x)=-2 \sin x\), \(g(x)=\sin 2x\), and \(h(x)= (f+g)(x)\) are all sinusoidal functions with different amplitudes and frequencies. The function \(h(x)= (f+g)(x)\) is obtained via pointwise addition of the y-values of f and g, resulting in a function that visually is a combination of these two.

Step 1: Sketch the graph of \(f(x)=-2 \sin x\)

This graph can be obtained by starting with the standard sine function, \(y=\sin x\), applying a vertical stretch by a factor of 2 and then a reflection in the x-axis. Since the standard sine function has a period of \(2 \pi\), the graph will features 2 full cycles one above and one below the x-axis in the interval \(0 \leq x \leq 2 \pi\)

Step 2: Sketch the graph of \(g(x)=\sin 2x\)

This graph can be obtained by starting with the standard sine function, \(y=\sin x\), and applying a horizontal compression by a factor of 2. This causes the period to decrease from \(2 \pi\) to \(\pi\). In the interval \(0 \leq x \leq 2 \pi\), we have 2 full cycles. Thus, compared to the standard sine function, it's moving twice as fast.

Step 3: Obtain the graph of \(h(x)= (f+g)(x)\)

To do this, for each x-coordinate within the given interval, calculate the corresponding y-coordinate by adding the y-coordinates of f and g. For example, at \(x=0\), \(f(0)=-2 sin(0)=0\), \(g(0)= sin(2*0)=0\), hence, \(h(0)=f(0)+g(0)=0+0=0\). Repeat this process for all x-coordinates to obtain the function \(h(x)\).

Key Concepts

Sine Function Transformations

Understanding the transformations of the sine function is integral to mastering trigonometry. The sine function, represented as \(y = \text{sin}(x)\), oscillates between -1 and 1, producing a wave-like pattern that repeats every \(2 \text{pi} (2 \pi)\) radians, known as the period.

Transformations involve shifting, stretching, compressing, or reflecting this wave. A transformation can be vertical (up or down), horizontal (left or right), involve a stretch or compression (making the wave taller or shorter, or squeezing or stretching it horizontally), or a reflection across the x-axis or y-axis (flipping it over).

These basic transformations can be combined to create complex waveforms, such as the ones described in the exercise, where we add or subtract the values of two transformed sine functions at each point to produce a new function.

Vertical Stretch

A vertical stretch takes the standard sine curve and elongates it along the y-axis. To perform a vertical stretch, we multiply the sine function by a factor greater than 1. For example, \(y = 2\text{sin}(x)\) indicates a vertical stretch of a factor of 2. This makes the peaks and troughs of the wave twice as high and low, respectively. The resulting graph retains the same period but differs in amplitude.

In the given exercise, the graph of \(f(x) = -2 \text{sin}(x)\) demonstrates a vertical stretch. Moreover, the negative sign indicates a reflection in the x-axis along with the stretch. Hence, not only is the graph taller, but it is also inverted when compared to the standard sine function.

Reflection in the X-Axis

Reflection in the x-axis is a transformation that inverts the graph of a function across the x-axis. Mathematically, if the original function is \(y = \text{sin}(x)\), its reflection is represented as \(y = -\text{sin}(x)\).

This reflection produces a mirror image of the shape of the function, but instead of peaks, we have troughs where the original function had its maximum values, and vice versa. The reflected graph will have the same period and frequency as the original. When we apply this transformation to the sine function, as with \(f(x) = -2 \text{sin}(x)\) from the exercise, the peaks and troughs flip, providing a fresh perspective on the function's behavior.

Horizontal Compression

Horizontal compression squeezes a function's graph closer together along the x-axis. It's achieved by multiplying the variable inside the function by a number greater than 1. In the context of the sine function, if we have \(y = \text{sin}(kx)\), a value of \(k > 1\) means the wave cycles complete more quickly, so the period is reduced.

For example, \(g(x) = \text{sin}(2x)\) in the exercise has a horizontal compression factor of 2, halving the function's period from \(2 \pi\) to \(\pi\), giving us two cycles within the interval \(0 \text{to} 2 \pi\). Hence, we see the sine wave completing its cycle in half the space on the x-axis.

Trigonometric Function Period

The period of a trigonometric function like the sine is the distance along the x-axis required for the function to complete one full cycle of its wave-like pattern. For the basic sine function, \(y = \text{sin}(x)\), the period is \(2 \pi\), meaning that the function repeats itself every \(2 \pi\) radians.

Transformations can alter this period. A horizontal stretch or compression, as seen in the exercise with \(g(x) = \text{sin}(2x)\), changes the period without affecting the range of the function. Having a solid understanding of how the period can change through transformations is crucial for graphing complex trigonometric functions and understanding their behavior.