Find the domain, range, amplitude, and period of y = -2 cos(-2x + π/4) - 1
Understand the Problem
The question is asking to determine specific characteristics of the cosine function provided, specifically its domain, range, amplitude, and period. This involves understanding the effects of the transformations applied to the basic cosine function.
Answer
Domain: $(-\infty, \infty)$, Amplitude: $2$, Period: $\pi$, Range: $[-3, 1]$
Answer for screen readers
- Domain: $(-\infty, \infty)$
- Amplitude: $2$
- Period: $\pi$
- Range: $[-3, 1]$
Steps to Solve
- Identify the Domain
The cosine function is defined for all real numbers. Therefore, the domain of $y = -2 \cos(-2x + \frac{\pi}{4}) - 1$ is: $$ \text{Domain} = (-\infty, \infty) $$
- Determine the Amplitude
The amplitude of a cosine function of the form $y = A \cos(Bx + C) + D$ is given by the absolute value of $A$. Here, $A = -2$, so the amplitude is: $$ \text{Amplitude} = |A| = |-2| = 2 $$
- Calculate the Period
The period of a cosine function is calculated using the formula: $$ \text{Period} = \frac{2\pi}{|B|} $$ For this function, $B = -2$, giving: $$ \text{Period} = \frac{2\pi}{|-2|} = \frac{2\pi}{2} = \pi $$
- Find the Range
The basic range of the cosine function is $[-1, 1]$. Considering the amplitude and vertical shift in our function, calculated as follows:
-
The maximum value occurs when $\cos$ is $1$: $$ y_{\text{max}} = -2(1) - 1 = -3 $$
-
The minimum value occurs when $\cos$ is $-1$: $$ y_{\text{min}} = -2(-1) - 1 = 1 $$
Thus, the range becomes: $$ \text{Range} = [-3, 1] $$
- Domain: $(-\infty, \infty)$
- Amplitude: $2$
- Period: $\pi$
- Range: $[-3, 1]$
More Information
The cosine function oscillates between its maximum and minimum values defined by its amplitude and vertical shift. The transformations applied, like the negative multiplier and addition, affect its orientation and position but not the existence of the function across all real numbers.
Tips
- Confusing the amplitude with the maximum or minimum value of the function.
- Neglecting the vertical shift when determining the range.
- Miscalculating the period by incorrectly identifying the coefficient of $x$.
AI-generated content may contain errors. Please verify critical information
