Graph a cosine function whose amplitude is 4, period is π , midline is y=−3 , and y-intercept is (0, 1) .
The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point. I appreciate the help thanks
To graph the given cosine function, let's break down the given information:
1. Amplitude: The amplitude of the cosine function is the maximum distance the graph reaches from the midline. Here, the given amplitude is 4.
2. Period: The period of a cosine function is the distance between any two consecutive maximum or minimum points on the graph. Here, the given period is π.
3. Midline: The midline is the horizontal line where the graph of the cosine function oscillates. The given midline is y = -3.
4. Y-intercept: The y-intercept of the graph is the point where it intersects the y-axis. Here, the given y-intercept is (0, 1).
Starting with the midline, we can plot the first point, which is (0, -3) since it falls on the midline.
Next, we need to determine the maximum or minimum point closest to the first point. Since the period is π, the graph completes one full oscillation between each consecutive maximum or minimum point. Therefore, we can determine the location of the second point by adding half of the period (π/2) to the x-coordinate of the first point.
To find the x-coordinate of the second point: x = 0 + (π/2) = π/2
Now, let us find the corresponding y-coordinate by using the cosine function:
cos(π/2) = 0
Therefore, the second point is (π/2, 0).
Now, we have the first two points (0, -3) and (π/2, 0) on the graph. Using this information, we can complete the graph of the cosine function.
Plotting the points and connecting them with a smooth curve, the graph of the cosine function with the given properties will look like a cosine wave oscillating above and below the midline (y = -3), with an amplitude of 4, a period of π, and passing through the y-intercept (0, 1).
I hope this helps!
To graph the cosine function with the given specifications, we need to use the general equation for cosine:
y = A * cos(Bx + C) + D
where:
A is the amplitude
B is the coefficient of x (1/period)
C is the phase shift
D is the vertical shift (midline)
From the given specifications:
Amplitude (A) = 4
Period (P) = π
Midline (D) = -3
The amplitude (A) represents the vertical stretch or compression of the graph, so it is 4. We can use this value to determine the maximum and minimum values of the graph.
The period (P) is the distance along the x-axis for one complete cycle of the graph. Since the given period is π, we can find the coefficient of x (B) using:
B = 1 / P = 1 / π
The midline (D) represents the vertical shift of the graph, so it is -3.
The phase shift (C) is not given, but since the first point must be on the midline, the phase shift is 0.
Using these values, the equation for the cosine function is:
y = 4 * cos((1/π)x + 0) - 3
To find the second point on the graph, we need to determine the maximum or minimum value closest to the first point. Since the first point is on the midline, the second point must be a maximum value.
A maximum value occurs when the cosine function is at its peak, which is when the angle inside the cosine function is equal to 0 or a multiple of 2π.
Since the period is π, the maximum value occurs at x = π/2.
Substituting x = π/2 into the equation, we can find the corresponding y-coordinate:
y = 4 * cos((1/π)(π/2) + 0) - 3
= 4 * cos(1/2) - 3
Using a calculator, we can find that cos(1/2) ≈ 0.8776.
So, the second point is approximately (π/2, 4 * 0.8776 - 3).
Now, we can summarize the information for the graph:
Amplitude (A) = 4
Period (P) = π
Midline (D) = -3
Phase Shift (C) = 0
First point on midline: (0, -3)
Second point closest to the first point: (π/2, 4 * 0.8776 - 3)
To graph the function, plot these two points and sketch the cosine curve between them. Remember, the cosine function repeats every period, so you can continue the pattern to the left and right of the graph.
I hope this helps! Let me know if you have any further questions.
midline: y = -3
y = -3 + cos(x)
amplitude=4:
y = -3 + 4cos(x)
period=π:
y = -3 + 4cos(2x)
y(0) = 1: works as-is
I expect you can find the points you want now. The graph is at
http://www.wolframalpha.com/input/?i=-3+%2B+4cos(2x)