How does the graph of y = –3cos (2θ + 45°) + 3 differ from the graph of y = cos (θ)
sifted right 22.5°
half the period
upside down
3 times as tall
shifted up 3 units
To understand how the graph of y = –3cos (2θ + 45°) + 3 differs from the graph of y = cos (θ), let's break it down step by step.
First, let's look at the equation y = cos (θ). This equation represents a standard cosine function, where y is a function of the angle θ. The graph of y = cos (θ) is a cosine wave that oscillates between -1 and 1. It starts at its maximum value of 1 when θ = 0, then moves through one complete cycle as θ increases, and finally returns to its maximum value at θ = 2π.
Now, let's examine the equation y = –3cos (2θ + 45°) + 3. In this equation, we have a few modifications compared to the previous one. Firstly, there is a negative sign (-) multiplying the cosine function. This negates the amplitude of the cosine wave, meaning the graph will be reflected in the x-axis. The standard cosine wave starts at its maximum value of 1, but now it will start at its minimum value of -1.
Additionally, the angle θ within the cosine function is multiplied by 2, which means the wave will complete two cycles within the same interval as the regular cosine function. This results in a faster oscillation.
Lastly, there is a phase shift of 45° (or π/4 radians) added to the angle θ inside the cosine function. This shifts the entire graph to the left by 45°.
Combining all these modifications together, the graph of y = –3cos (2θ + 45°) + 3 has a reflected and compressed wave compared to y = cos (θ). It starts at its minimum value of -4 and oscillates between -4 and -2. Additionally, the graph is shifted to the left by 45°.
Remember, when encountering similar problems, you can analyze the equation by focusing on the key components such as amplitude, reflection, compression/stretch, phase shift, and horizontal/vertical translations to determine how the graph differs from the standard function.