State the period, amplitude, max/min values, range, domain, horizontal phase shift and vertical displacement.
y = -3cos (2x - 30°) + 1
-3: amplitude = 3
cos (2x - 30°): period = 360/2 = 180°
+1: vertical shift = up 1
max/min of cos are ±1, so with our amplitude and vertical displacement, max/min = (0+1)±3 = 4,-2
range is [min,max] = [-2,4]
domain is all real numbers, as always
cos(2x-30°) = cos(2(x-15°)) so horizontal displacement is 15° to the right
To identify the period, amplitude, max/min values, range, domain, horizontal phase shift, and vertical displacement of the function y = -3cos (2x - 30°) + 1, we can analyze the equation.
1. Period:
The period of a cosine function is determined by the coefficient in front of the x variable. In this case, the coefficient is 2, so the period can be calculated using the formula:
Period = 2π / coefficient = 2π / 2 = π
2. Amplitude:
The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine function. In this case, the coefficient is -3. The amplitude is always positive, so the amplitude is | -3 | = 3.
3. Maximum and Minimum Values:
The maximum and minimum values of a cosine function with a vertical displacement can be calculated by adding or subtracting the amplitude from the vertical displacement. In this case, the vertical displacement is +1. Therefore, the maximum value would be 3 + 1 = 4, and the minimum value would be -3 + 1 = -2.
4. Range:
The range of the function is the set of all possible output values. For this cosine function, the range is the interval from the minimum value (-2) to the maximum value (4). Therefore, the range is [-2, 4].
5. Domain:
The domain of the function is the set of all possible input values. Since cosine functions have a periodic nature, the domain is typically all real numbers. So, the domain is (-∞, +∞).
6. Horizontal Phase Shift:
The horizontal phase shift of a cosine function is given by the value inside the parentheses. In this case, the shift is -30°. The phase shift is the opposite of the sign, so the horizontal phase shift is +30°.
7. Vertical Displacement:
The vertical displacement of the function is the constant added or subtracted to the cosine function. In this case, the vertical displacement is +1.
In summary:
- Period = π
- Amplitude = 3
- Maximum/Minimum Values = 4 / -2
- Range = [-2, 4]
- Domain = (-∞, +∞)
- Horizontal Phase Shift = +30°
- Vertical Displacement = +1
To find the period, amplitude, max/min values, range, domain, horizontal phase shift, and vertical displacement of the given function, we will use the general form of a cosine function:
y = A*cos(B(x - C)) + D
The given function can be rewritten in this form as:
y = -3*cos(2(x - 15°)) + 1
Now, let's break down the parts:
1. Period:
The period of a cosine function is given by the formula 2π/B. In this case, B = 2, so the period is 2π/2 = π.
2. Amplitude:
The amplitude of a cosine function is the absolute value of the coefficient A. In this case, A = -3, so the amplitude is |-3| = 3.
3. Max/Min values:
For a cosine function, the maximum value is equal to the amplitude + vertical displacement, and the minimum value is equal to the negative amplitude + vertical displacement. In this case, the maximum value is 3 + 1 = 4, and the minimum value is -3 + 1 = -2.
4. Range:
The range of a cosine function is the set of all possible y-values. In this case, the range is [-2, 4].
5. Domain:
The domain of a cosine function is the set of all possible x-values. Since cosine functions repeat every period, the domain is all real numbers.
6. Horizontal phase shift:
The horizontal phase shift is given by C°, which represents a shift in the x-direction. In this case, C = 15°, so there is a horizontal phase shift of 15° to the right.
7. Vertical displacement:
The vertical displacement is given by D, which represents a shift in the y-direction. In this case, D = 1, so there is a vertical displacement of 1 unit upwards.
Summary:
- Period: π
- Amplitude: 3
- Maximum value: 4
- Minimum value: -2
- Range: [-2, 4]
- Domain: all real numbers
- Horizontal phase shift: 15° to the right
- Vertical displacement: 1 unit upwards
y = -3cos (2x - 30°) + 1
y = -3cos (2(x - 15°)) + 1
compare this y = a cos(k(𝝷 + d) + c
where
a is the amplitude
period = 2π/k
d is a horizontal phase shift of d units to the left ( for -d , the shift is to the right)
c is the vertical displacement.
for the max/min values, range, domain, you should be able to find these from the above information.
Let me know what your answers are.