Find the domain, range, amplitude, and period of y = -2 sin(3x + π).
Understand the Problem
The question is asking to determine specific characteristics of the sine function represented by the equation y = -2 sin(3x + π). This includes finding the domain, range, amplitude, and period of the function.
Answer
Domain: $(-\infty, \infty)$, Amplitude: $2$, Period: $\frac{2\pi}{3}$, Range: $[-2, 2]$.
Answer for screen readers
- Domain: $(-\infty, \infty)$
- Amplitude: $2$
- Period: $\frac{2\pi}{3}$
- Range: $[-2, 2]$
Steps to Solve
- Finding the Domain
The sine function is defined for all real numbers. Therefore, the domain of $y = -2 \sin(3x + \pi)$ is:
$$ \text{Domain: } (-\infty, \infty) $$
- Finding the Amplitude
The amplitude of a sine function is determined by the coefficient in front of the sine. Here, the coefficient is $-2$. The amplitude is always a positive value, so:
$$ \text{Amplitude: } | -2 | = 2 $$
- Finding the Period
The period of the function can be determined from the coefficient of $x$ inside the sine function. The formula for the period $T$ is given by:
$$ T = \frac{2\pi}{|b|} $$
where $b$ is the coefficient of $x$. For this function, $b = 3$, thus:
$$ T = \frac{2\pi}{3} $$
- Finding the Range
The range of the sine function is affected by both the amplitude and the vertical shift (which is 0 here). The range can be calculated as follows:
$$ \text{Range: } [-\text{Amplitude}, \text{Amplitude}] $$
Since the amplitude is 2, we get:
$$ \text{Range: } [-2, 2] $$
- Domain: $(-\infty, \infty)$
- Amplitude: $2$
- Period: $\frac{2\pi}{3}$
- Range: $[-2, 2]$
More Information
The sine function oscillates between -1 and 1. The coefficient alters the height (amplitude) and reflects it across the x-axis if negative. The period indicates how often the function repeats.
Tips
- Misinterpreting the amplitude as negative; amplitude should always be positive.
- Confusing the range with amplitude; ensure the vertical shift is considered.
- Incorrectly calculating the period by forgetting to use the absolute value of the coefficient.
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