what is the range of each of these equations?
1. 4sin(3x-1)+4
2. -4sin(-3x+1)-4
3. 2sin(1.5x+.5)+2
4. -2sin(-1.5x-.5)-2
To find the range of each equation, we need to determine the minimum and maximum values that the equations can take.
1. Equation: 4sin(3x-1)+4
To find the range, we need to find the maximum and minimum values of the function sin(3x-1). The range of the sine function is -1 to 1. Since we multiply the sine function by 4 and add 4, the final range will be shifted up by adding 4 to the minimum and maximum values.
Minimum value: -1 * 4 + 4 = 3
Maximum value: 1 * 4 + 4 = 8
So, the range of the first equation is [3, 8].
2. Equation: -4sin(-3x+1)-4
Similar to the first equation, we need to consider the range of sin(-3x+1). The range of the sine function is -1 to 1, and since we multiply it by -4 and subtract 4, the final range will be shifted down.
Minimum value: -1 * -4 - 4 = 0
Maximum value: 1 * -4 - 4 = -8
So, the range of the second equation is [-8, 0].
3. Equation: 2sin(1.5x+0.5)+2
Again, we need to consider the range of sin(1.5x+0.5). The range of the sine function is -1 to 1, and by multiplying it by 2 and adding 2, the final range will be shifted up.
Minimum value: -1 * 2 + 2 = 0
Maximum value: 1 * 2 + 2 = 4
So, the range of the third equation is [0, 4].
4. Equation: -2sin(-1.5x-0.5)-2
Similarly, we consider the range of sin(-1.5x-0.5). The range of the sine function is -1 to 1, and by multiplying it by -2 and subtracting 2, the final range will be shifted down.
Minimum value: -1 * -2 - 2 = 0
Maximum value: 1 * -2 - 2 = -4
So, the range of the fourth equation is [-4, 0].
Therefore, the ranges for the given equations are:
1. [3, 8]
2. [-8, 0]
3. [0, 4]
4. [-4, 0]