I am doing a similar assignment as Anaa please help!
y = 3cos(x-1)
Amplitude:
Period:
Two x - intercepts:
y-intercept:
Domain:
Range:
Shift along x:
Shift along y:
Kind of reflection
Thank you so much!
Sure! I'd be happy to help you with your assignment. Let's break down the given equation and find the answers to each question:
The equation provided is: y = 3cos(x-1)
Amplitude:
To find the amplitude of the cosine function, we need to look at the coefficient of the cosine part. In this case, the coefficient is 3. The absolute value of this coefficient represents the amplitude. So, the amplitude of the given function is |3| = 3.
Period:
The period of a cosine function can be found using the formula: T = 2π / b, where b is the coefficient multiplying x inside the cosine function. In this case, the coefficient b is 1. Therefore, the period of the given function is T = 2π / 1 = 2π.
Two x-intercepts:
To find the x-intercepts, we need to solve the equation for y = 0. So, we set y = 0 and solve the equation: 0 = 3cos(x-1).
To find the solutions, we know that cosine is zero at 90° (or π/2 radians) intervals. Therefore, x-1 = π/2 or x-1 = 3π/2.
Solving for x, we get x = π/2 + 1 and x = 3π/2 + 1.
Therefore, the two x-intercepts of the given function are x = π/2 + 1 and x = 3π/2 + 1.
y-intercept:
The y-intercept is the value of y when x is 0. So, we plug in x = 0 into the equation:
y = 3cos(0-1)
y = 3cos(-1)
Evaluating the cosine of -1, we get y = 3 * 0.5403 ≈ 1.6209.
Therefore, the y-intercept of the given function is approximately (0, 1.6209).
Domain:
The domain of a cosine function is all real numbers. So, the domain of the given function is (-∞, +∞).
Range:
The range of a cosine function is between -1 and 1 (inclusive). Since the amplitude of the given function is 3, the range will be between -3 and 3 (inclusive). Therefore, the range of the given function is [-3, 3].
Shift along x:
The function has a shift of 1 unit to the right along the x-axis. This is because the equation is cos(x-1) rather than cos(x). Therefore, the function is shifted right by 1 unit.
Shift along y:
The function does not have a shift along the y-axis. This is because there is no constant term added or subtracted in the equation.
Kind of reflection:
The cosine function is an even function, which means it is symmetric about the y-axis. Therefore, there is no reflection in this case.
I hope this helps you with your assignment! Let me know if you have any further questions.