Question

Sketch the graph of the curve that results after applying each transformation to the graph of the function \(f(x)=\sin x\). a) \(f\left(x-\frac{\pi}{3}\right)\) b) \(f\left(x+\frac{\pi}{4}\right)\) c) \(f(x)+3\) d) \(f(x)-4\)

Short answer

Horizontally shift by \( \frac{\pi }{3} \) right and \( \frac{\pi }{4} \) left, then vertically shift by 3 units up and 4 units down.

Step-by-step solution

Identify the Base Function

The base function is given as \(f(x) = \sin x\). This is a standard sine function which oscillates between -1 and 1 with a period of \(2\pi\).

Apply Horizontal Shift for Part (a)

For \(f\left(x-\frac{\pi}{3}\right)\), the function is horizontally shifted to the right by \(\frac{\pi}{3}\). This transformation does not affect the amplitude or the period of the function.

Apply Horizontal Shift for Part (b)

For \(f\left(x+\frac{\pi}{4}\right)\), the function is horizontally shifted to the left by \(\frac{\pi}{4}\). Again, the amplitude and the period remain the same.

Apply Vertical Shift for Part (c)

For \(f(x)+3\), the entire function is shifted vertically upwards by 3 units. The new maximum value becomes 4 and the minimum value becomes 2.

Apply Vertical Shift for Part (d)

For \(f(x)-4\), the entire function is shifted vertically downwards by 4 units. The new maximum value becomes -3 and the minimum value becomes -5.

Combine Information for Sketching

Using the shifts described above, sketch the graphs of each transformed function. Ensure the horizontal shifts for (a) and (b) and the vertical shifts for (c) and (d) are accurately represented on the graph.

Key concepts

Sine Function

Let's start with the basics. The sine function is written as \( f(x) = \sin x \), and it is one of the fundamental trigonometric functions. This function produces a smooth, wave-like pattern that oscillates between -1 and 1. The zeros of the sine function occur at multiples of \pi (0, \pi, 2\pi, ...) -------- signifying the points where the curve crosses the x-axis.
This function is periodic, meaning it repeats its pattern every \( 2\pi \) units. This repeating pattern is what gives trigonometric functions their unique properties. The sine function starts at zero, rises to a maximum value of 1, falls back to zero, decreases to a minimum value of -1, and finally returns to zero to complete one full cycle.

Due to its simple yet flexible form, the sine function is often used as the base function for various transformations, including horizontal shifts, vertical shifts, and more complex modifications. Understanding the basic properties of the sine function helps greatly in graph transformations.

Horizontal Shift

A horizontal shift involves moving the entire function left or right along the x-axis. In our examples, we have two horizontal shifts to consider:
When we shift the function to the right, as in \( f\left( x - \frac{ \pi }{ 3 } \right) \), we move each point on the graph right by \( \frac{ \pi }{ 3 } \) units. This doesn't change the function's amplitude (height) or period (distance required for one complete cycle). Only the position of the sine curve on the x-axis is modified. Essentially, this transformation means that peaks, troughs, and zeros of the sine curve will occur later than they used to.

For a left shift, such as \( f\left( x + \frac{ \pi }{ 4 } \right) \), each point on the original graph is moved to the left by \( \frac{ \pi }{ 4 } \) units. This also maintains the amplitude and period but shifts features like peaks and valleys to the left.

In both cases, it's crucial to note that the shape of the sine wave remains unchanged. Only its horizontal positioning is altered.

Vertical Shift

In contrast to horizontal shifts, vertical shifts involve moving the graph up or down along the y-axis. This is done by adding or subtracting a constant value from the sine function. Let’s see how we deal with this:
For an upward shift, like \( f(x) + 3 \), we move every point on the graph up by 3 units. This affects both the maximum and minimum values of the function. The peak of the sine wave, originally at 1, moves to 4, and the trough, originally at -1, shifts to 2. Essentially, we add 3 to every y-coordinate on the graph.

On the other hand, a downward shift, such as \( f(x) - 4 \), moves every point down by 4 units. This results in new maximum and minimum values of -3 and -5, respectively. Similar to the upward shift, we subtract 4 from every y-coordinate on the graph.

These vertical shifts affect where the sine function oscillates around the y-axis but don't change the period or the shape of the sine wave.

Periodic Functions

A key property of the sine function is that it's a periodic function. This means it repeats its pattern at regular intervals. For the sine function, this interval is \( 2\pi \). Every \( 2\pi \) units along the x-axis, the sine function completes one full cycle and starts anew.
This periodic nature is what gives the sine function and other trigonometric functions their oscillating wave forms. Periodic functions are essential in various applications, from signal processing to describing wave phenomena in physics.

When we apply transformations like horizontal and vertical shifts, we modify where each cycle of the function starts and ends or moves the entire wave up or down. Yet the periodic nature remains unaltered unless explicitly changed (e.g., by altering the function's frequency). Understanding periodicity helps predict and analyze wave patterns and transformations.

In summary, the sine function's periodicity, combined with horizontal and vertical shifts, creates a versatile tool for modeling and understanding wave-like behaviors in different contexts.