To determine the graph of the function \( f(x) = 2 \cos\left(x + \frac{\pi}{3}\right) - 1 \), let's analyze the function step by step.
-
Amplitude and Vertical Shift:
- The amplitude of the cosine function is the coefficient in front of the cosine, which is 2. Thus, the graph will oscillate between \( -1 \) (minimum) and \( 2 \) (maximum).
- There is a vertical shift downward by 1 unit. This makes the range of the function from \( -1 - 1 = -2 \) (minimum) to \( 2 - 1 = 1 \) (maximum). Thus, the function oscillates between \(-1\) and \(1\).
-
Phase Shift:
- The cosine function has a phase shift of \( -\frac{\pi}{3} \) since the function is \( \cos(x + \frac{\pi}{3})\). This phase shift will shift the entire graph left by \( \frac{\pi}{3} \).
-
Key Points:
- At \( x = -\frac{\pi}{3} \), the function reaches its maximum value \( f\left(-\frac{\pi}{3}\right) = 2 \cos(0) - 1 = 2(1) - 1 = 1 \).
- At \( x = 0 \): \( f(0) = 2 \cos\left(\frac{\pi}{3}\right) - 1 = 2\left(\frac{1}{2}\right) - 1 = 1 - 1 = 0 \).
- At \( x = \frac{\pi}{3} \): \( f\left(\frac{\pi}{3}\right) = 2 \cos\left(\frac{2\pi}{3}\right) - 1 = 2 \left(-\frac{1}{2}\right) - 1 = -1 - 1 = -2 \).
-
Behavior of the Graph:
- The function will reach its maximum at \(-\frac{\pi}{3}\), there will be a value of zero at \(0\), and it will reach its minimum at \(x = \frac{\pi}{3}\).
Now, we can analyze the given options based on the key points and characteristics we found:
- The first response includes points that are consistent with the expected behavior of the graph.
- The fourth response mentions \( (-\frac{\pi}{3}, 3) \) which is inconsistent with our findings as the maximum is \(1\).
- The second and third responses don't match the expected behavior regarding maximum and minimum values.
Based on this analysis, the correct graph corresponds to the first response:
A plane curve is graphed. The curve passes through:
- \( \left(-\frac{2\pi}{3}, -1\right) \)
- \( \left(-\frac{\pi}{3}, 0\right) \)
- \( \left(0, 2\right) \)
- \( \left(\frac{\pi}{3}, 3\right) \)
- \( \left(\pi, 0\right) \).
So the answer is: The graph of \( f(x) = 2 \cos\left(x + \frac{\pi}{3}\right) - 1 \) is the first option.
1 answer