Problem 14 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=2 \sin x\)

Step-by-Step Solution

Answer

The amplitude of the function \(g(x) = 2 \sin x\) is 2 and the period is \(2\pi\). The function is a vertical stretch (by a factor of 2) of the parent function \(f(x) = \sin x\) with no other transformations.

Step 1: Identifying amplitude and Period

Recall that the standard form of a sinusoidal function is \(f(x) = A \sin(Bx + C) + D\) where:\n A is the amplitude (which is always positive),\n B determines the period of the function, which is \( \frac{2\pi}{|B|}\),\n C shifts the graph horizontally, and \n D moves the graph vertically.\n\nIn the given function \(g(x) = 2 \sin x\), A is 2 which means the amplitude, is 2. There are no B, C, and D values specified, so B is 1, which results with a period of \( \frac{2\pi}{|1|} = 2\pi\). So, amplitude is 2 and period is \(2\pi\).

Step 2: Graphing the function and Describing transformation

Since D is not specified in the given function, there is no vertical shift (the graph remains centred on the x-axis). Similarly, with no C value, there is no horizontal shift either. As B was not specified, it causes no stretching or compressing. The only transformation that occurs is due to A being twice as large as it is in the function \(f(x) = sinx\), the parent of \(g(x)\). This causes a vertical stretch. We can graph this function by first drawing the standard sine curve and then transforming it according to these observations: stretching it vertically by a factor of 2. The period of the function is \(2\pi\). Plot 'x' and 'y' keeping the period, amplitude, and transformation in mind.

Key Concepts

Amplitude

Amplitude is a measure of how much a wave varies from its mean position. In the context of trigonometric functions like sine and cosine, the amplitude dictates how far the crest and trough of the wave are from the central axis (usually the x-axis).

For the function \(g(x) = 2 \sin x\), the amplitude is the absolute value of the coefficient in front of the sine function. Here, it is 2. This tells us that the peaks of the sine wave reach up to 2 on the y-axis and the troughs reach down to -2. Simply put, the sine wave is stretched vertically.

Sine functions, when undistorted, have an amplitude of 1 (peaks at 1 and troughs at -1). Any variation of this is due to multiplying the sine by a factor. It's important to remember that amplitude is always positive, and reflectivity (flipping the wave over the x-axis) does not change this if the sign is negative.

Period

The period of a trigonometric function is the length of one complete cycle on the graph. For sine and cosine functions, this cycle represents the point where the function begins to repeat its values. Knowing this helps us predict the graph's behavior.

The formula to find the period of a sine or cosine function is \( \frac{2\pi}{|B|} \), where \(B\) is the coefficient of \(x\) in the function \(f(x) = A \sin(Bx + C) + D\). In our function \(g(x) = 2 \sin x\), \(B\) is 1 (since there is no coefficient displayed). This means the period remains \(2\pi\). This tells us that every \(2\pi\) units on the x-axis, the sine function starts anew.

The concept of period is essential for graphing because it defines where our function will repeat, showing its predictable pattern over time. Without changes in \(B\), the typical sine wave maintains its traditional period.

Graph Transformation

Graph transformations involve shifting, stretching, shrinking, or reflecting the graph of a function. For sinusoidal functions, these transformations are determined by the values of \(A\), \(B\), \(C\), and \(D\) in the standard form.

In the function \(g(x) = 2 \sin x\), the primary transformation is a vertical stretch caused by \(A = 2\).

• The absence of any \(C\) value implies no horizontal shift (or phase shift), keeping our sine wave aligned with the y-axis.
• The lack of a \(D\) value means there is no vertical translation; the function oscillates symmetrically around the origin.
• Since \(A\) is 2, each point on the curve is twice as far from the x-axis compared to the parent function \(\sin x\).

Understanding these basics makes it simpler to graph the transformation. Start with the basic sine graph, apply the vertical stretch, and visualize how this manipulation affects the function's oscillation and reach.