Problem 31 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=\sin x\) by basic transformations: (a) \(\sin (\pi x)\) (b) \(\sin \left(x+\frac{\pi}{4}\right)\) (c) \(-2 \sin (\pi x+1)\)

Step-by-Step Solution

Answer

(a) Horizontal compression; (b) Left shift; (c) Horizontal compression, left shift, vertical stretch and reflection.

Step 1: Understanding the Parent Function

The parent function given is \( y = \sin x \). This function represents a sinusoidal wave where the amplitude is 1, the period is \( 2\pi \), and there is no phase shift or vertical translation.

Step 2: Transformation for \( y = \sin(\pi x) \)

The function \( y = \sin(\pi x) \) involves a horizontal compression. The standard period \( 2\pi \) of the sine wave gets altered. To find the period, divide the original period by the coefficient of \( x \), which is \( \pi \). So, the new period is \( \frac{2\pi}{\pi} = 2 \). This transformation compresses the wave horizontally.

Step 3: Phase Shift for \( y = \sin\left(x + \frac{\pi}{4}\right) \)

The function \( y = \sin\left(x + \frac{\pi}{4}\right) \) is a phase shift to the left. Since the sine function is of the form \( y = \sin(x - c) \) causing a shift to the right by \( c \) units, in this case, adding \( \frac{\pi}{4} \) causes a shift to the left by \( \frac{\pi}{4} \) units.

Step 4: Multiple Transformations for \( y = -2 \sin(\pi x + 1) \)

The function \( y = -2 \sin(\pi x + 1) \) involves several transformations. First, the \( \pi x \) inside the sine causes a horizontal compression with a period of 2 (similar to Step 2), and adding 1 inside the function \( \pi x + 1 \) shifts the wave to the left by \( \frac{1}{\pi} \) units. Moreover, the coefficient \(-2\) outside the function indicates a vertical stretch by a factor of 2 and a reflection over the x-axis.

Key Concepts

Understanding Horizontal Compression in Trigonometric Functions

Horizontal compression occurs when the period of a trigonometric function like the sine function is reduced. For the sine function, the standard period is \(2\pi\). However, when you transform this function to \(y = \sin(\pi x)\), something interesting happens. Here, the coefficient \(\pi\) multiplies the angle \(x\). This effectively compresses the period of the sine wave.

The new period is calculated by taking the original period, \(2\pi\), and dividing it by the coefficient of \(x\), which is \(\pi\). Therefore, you get \(\frac{2\pi}{\pi} = 2\). In simpler terms, the entire wave, which used to repeat every \(2\pi\) units, now completes its cycle every 2 units on the x-axis.

• This compression means the wave is tighter, completing its ups and downs faster.
• The number of complete cycles in a given interval increases.
• Each cycle is more frequent but occurs over less of the horizontal space.

Understanding Phase Shift in Trigonometric Functions

A phase shift refers to the translation of a wave along the horizontal axis. When examining the function \(y = \sin\left(x + \frac{\pi}{4}\right)\), a phase shift occurs. This function adds \(\frac{\pi}{4}\) to the variable \(x\). Instead of starting at the origin, the wave starts earlier — essentially, we see a shift to the left.

In general terms, if a function \(y = \sin(x - c)\) shifts to the right by \(c\) units, adding inside like \(x + c\) shifts it to the left by \(c\) units.

• This is due to the fact that the wave 'reaches' its starting point sooner.
• In sinusoidal patterns, it mimics the cyclical movement occurring earlier than usual.
• This transformation doesn't change the shape or the amplitude of the wave; it only repositions it along the x-axis.

Decoding Vertical Stretch in Trigonometric Functions

A vertical stretch involves amplifying the height of the wave without altering its period or horizontal placement. In the function \(y = -2\sin(\pi x + 1)\), the coefficient \(-2\) plays a crucial role.

This coefficient outside the sine function causes the wave to stretch vertically by a factor of 2. The amplitude, which is normally 1 for \(y = \sin x\), becomes 2. This means that each peak and trough of the wave is twice as high and low as usual. The negative sign in front of the 2 indicates a reflection over the x-axis as well, inverting the wave.

• Vertical stretch causes the peaks and troughs to extend further from the x-axis.
• The reflection makes all the peaks become troughs and vice versa.
• This transformation predominantly affects the amplitude, changing how "tall" the wave appears.