Graphing trigonometric functions
A sinusoidal function has a maximum at (2,10) and its next minimum (5,-2) find an equation that represents this situation
A sinuosoidal function has a zero at (5,0) and its next minimum is (7,-4). find an equation that represents this situation.
I really don't get how to come up with an equation, this chapter was really the hardest for me. I don't get how to find amplitude, vertical displacement, phase shift
Can someone please explain step by step
Notice the max is 10 and the min is -2, for a range of 12
which makes the amplitude = 6
the axis would be 4 units high
so far we have something like
y = 6 sin (.......) + 4
the distance from the max to the min along the horizontal is 3 (from 2 to 5 is 3 units)
but the period would be twice that
so the period is 6
remember 2π/k = period
2π/k = 6
6k = 2π
k = 2π/6 = π/3
Ok, now also have the period
and our equation must look something like this:
y = 6 sin (π/3)(x + d) + 4
almost done....
we just have to sub in one of our points to find d , the phase shift
let's use (2,10)
10 = 6sin π/3(2 - d) + 4
1 = sin π/3(2-d)
I know that sin π/2 = 1
so π/3(2-d) = π/2
divide both sides by π
(1/3)(2-d) = 1/2
times 6
2(2-d) = 3
2-d = 3/2
-d = 3/2 - 2 = -1/2
d = 1/2
y = 6 sin (π/3)(x + 1/2) + 4
check for the other point:
if x = 5, we should get y = -2
y = 6 sin (π/3)(5+1/2) + 4
= 6 sin ((π/3)(11/2)) + 4
= 6 sin (11π/2) + 4
= 6(-1) + 4 = -2 , YEAHHHHH
I chose to use a sine curve for no specific reason
We could have used a cosine as well.
The period and amplitude and vertical shift would be the same
the change would be in the value of d
Try the second one, it is slightly different.
Make a sketch to see what you are doing.
so you added 10 and 2 to get 12, and amplitude is the middle of max and min?
how did you get the axis 4 units?
a period is always twice of something?
the phase shift is sort of confusing me, because isnt that just between the x values? 2, and 5? is there some other way to figure it out?
"how did you get the axis 4 units? "
where is the middle value between -2 and 10
sketch on a number line to see.
the amplitude, or the a value, is 1/2 the difference between the max and the min
difference = 10 - (-2) = 12
half of that is 6
I illustrated with detailed calculations, how I got the phase shift d
Graphing trigonometric functions involves understanding key properties such as amplitude, vertical displacement, and phase shift. Let's break down the process step by step for both situations:
Situation 1: Maximum and Minimum Points Given
Step 1: Find the Amplitude
The amplitude of a sinusoidal function is the distance between the maximum/minimum value and the horizontal axis. In this case, the maximum value is 10 and the next minimum value is -2. The amplitude is the absolute value of the difference between these two values: |10 - (-2)| = 12.
Step 2: Determine the Vertical Displacement
The vertical displacement is the average of the maximum and minimum values. In this situation, the maximum is 10 and the minimum is -2. The vertical displacement is the average of these two values: (10 + (-2))/2 = 4.
Step 3: Find the Phase Shift
The phase shift is the horizontal shift of the sinusoidal function. In this situation, the maximum occurs at x = 2, which is halfway between the maximum and the minimum. Thus, the phase shift is 2 units to the right.
Step 4: Determine the Equation
The general form of a sinusoidal function is y = A*sin(B(x - C)) + D.
- The amplitude (A) in this case is 12.
- The vertical displacement (D) is 4.
- The phase shift (C) is 2.
- B is the frequency coefficient. Since it is not mentioned, we assume B = 1.
Therefore, the equation that represents this situation is y = 12*sin(x - 2) + 4.
Situation 2: Zero and Minimum Points Given
Step 1: Find the Amplitude
Similar to the previous situation, the amplitude is the distance between the maximum/minimum value and the horizontal axis. However, in this case, we only have the zero at (5,0), so the amplitude is not explicitly given. We assume that the minimum value is the opposite of the amplitude. Thus, the minimum value is -4.
Step 2: Determine the Vertical Displacement
The vertical displacement is the average of the maximum and minimum values. In this case, the minimum is -4, and the average value can be calculated as (-4 + 0)/2 = -2.
Step 3: Find the Phase Shift
The phase shift is not explicitly given in this situation. However, we know that the zero occurs at x = 5 and the next minimum is at x = 7. Thus, the phase shift can be determined by finding the difference between these two x-coordinates: 7 - 5 = 2 units to the right.
Step 4: Determine the Equation
Using the same general form y = A*sin(B(x - C)) + D:
- The amplitude (A) is the distance between 0 and -4, which is 4.
- The vertical displacement (D) is -2.
- The phase shift (C) is 2.
- Assumed B = 1.
Therefore, the equation that represents this situation is y = 4*sin(x - 2) - 2.
Remember, these equations are specific to the given situations. The process may vary depending on the information provided and the problem at hand.