Question

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\sin x, g(x)=\sin 3(x+3 \pi)-5\)

Short answer

The graph of the function \(g(x)=\sin 3(x+3 \pi)-5\) compared to \(f(x)=\sin x\) has a frequency of 3 causing it to oscillate three times as fast. Also, the graph has a phase shift of \(3 \pi\) to the left and is shifted 5 units down.

Step-by-step solution

Identify the Frequency Change

The coefficient in front of \(x\) in the sine function determines the frequency of the graph. In the function \(f(x)=\sin x\), the frequency is 1, but in the function \(g(x)=\sin 3(x+3 \pi)-5\), the frequency is 3. Therefore, the graph of \(g\) will oscillate three times as fast as the graph of \(f\).

Identify the Phase Shift

The phase shift is determined by the value added or subtracted from \(x\) inside the sine function. In this case, \(x+3 \pi\) is used inside the sine function of \(g(x)\), so there is a phase shift of \(3 \pi\) to the left compared to the function \(f(x)=\sin x\) which has no phase shift.

Identify the Vertical Shift

The subtraction of 5 in the function \(g(x)=\sin 3(x+3 \pi)-5\) indicates that the graph is shifted down by 5 units compared to the graph of \(f(x)=\sin x\).

Key concepts

Frequency Change

The concept of frequency change in sine functions revolves around how often the wave pattern repeats itself over a certain interval. In the context of trigonometric functions, this frequency change is exhibited by the coefficient before the variable within the sine function.

For example, with our original sine function, which is denoted as \( f(x)=\text{sin} x \), the usual pattern completes one full cycle from 0 to \( 2\pi \) radians. However, when we introduce a coefficient, such as 3 in the transformation to \( g(x)=\text{sin} 3(x+3\pi) - 5 \), the frequency of the sine wave is altered. Now, the sine function completes three cycles within the same interval that the original function completed just one. This means that the graph of \( g(x) \) will have three peaks and three troughs between 0 and \( 2\pi \), making the function oscillate three times faster than the original sine wave.

Phase Shift

A phase shift in trigonometric functions, specifically the sine function, can be thought of as a horizontal translation of the entire graph. This shift occurs when a value is either added or subtracted from the variable 'x' within the sine function, resulting in the graph 'moving' along the x-axis.

In the given function \( g(x)=\text{sin} 3(x+3\pi) - 5 \), we witness a phase shift because of the addition of \( 3\pi \) to 'x'. This addition means that every point on the graph of \( f(x)=\text{sin} x \) is moved to the left by an angle of \( 3\pi \) radians before undergoing the frequency change. Visualizing this can be quite insightful: you can imagine taking the sine curve and sliding it to the left. The amount it slides is what we call the phase shift, which, in this case, is a sizable shift given that \( 3\pi \) represents 1.5 full cycles of the standard sine wave.

Vertical Shift

Moving onto vertical shifts, we describe a change in the 'height' of a trigonometric function’s graph on the Cartesian plane. A vertical shift earmarks a uniform uplifting or down-pressing of the entire graph, dictated by the constant that is added to or subtracted from the function.

For the function at hand, \( g(x)=\text{sin} 3(x+3\pi) - 5 \), the '-5' at the end of the expression influences a downward shift of the graph. This means the usual midline of the sine wave at 'y = 0' is now repositioned to 'y = -5'. The peak and trough points follow suit, moving down by 5 units. Consequently, where the original sine wave oscillates around the x-axis, our transformed function oscillates around the line 'y = -5'. Whenever facing vertical shifts, think of the entire graph as being nudged up or down along the y-axis by the given value.

Trigonometric Functions

At the heart of our topic are trigonometric functions—sine, cosine, tangent and their reciprocals—each with unique properties and graphs that model periodic phenomena. The sine function in particular, represented as \( y = \text{sin} x \), showcases a smooth, continuous wave that oscillates between 1 and -1, representing the y-coordinate of a point rotating on the unit circle.

Understanding trigonometric functions is invaluable, as they are foundational in describing and modeling various physical phenomena, such as sound waves, light waves, and other types of oscillations. When discussing transformations, it’s important to realize that they don't alter the fundamental shape of the sine curve; they simply reposition it. Each transformation, whether it’s a change in frequency, a phase shift, or a vertical shift, gives us a new way to represent complex oscillations in a way that is tailored to the physical context we are analyzing.