find the period, amplitude,zeros, extreme points at maximum and minimum of the given function:
1.y = sin( -2x)
2.y = 4 sin( 5x)
3.y = -3 sin (3x/5)
4.y = 1/2 cos (-4x)
your equations all fit the pattern
y = a (sin or cos)(kx)
where |a| is the amplitude and the period is 2π/k
I will do one of them , you do the others in the same way.
2.
y = 4sin(5x)
amplitude is 4
period = 2π/5 radians
dy/dx = 4cos(5x) (5) = 20 cos(5x)
= 0 for any max/min
20cos(5x) = 0
cos(5x) = 0
I know cos(π/2) = 0 and cos(3π/2) = 0
so 5x = π/2 or 5x = 3π/2
x = π/10 or x = 3π/10
when x = π/10 , y = 4sin(5(π/10)) = 4sin(π/2) = 4
when x = 3π/10 y = 4sin(5(3π/10)=4sin(3π/2) = -4
So in the first period, the max point is (π/10 , 4)
and the min point is (3π/10 , -4)
I used Calculus to find the max/min.
We could have done this by simply knowing something about our curve.
We know we have a sine curve of period 2π/5
So mark off a section of the line from 0 to 2π/5
and marking its 4 quadrants as
0 π/10 2π/10 3π/10 and 4π/10
or
0 , π/10 , π/5 , 3π/10 , and 2π/5
We know this sine curve starts at 0 goes up to 4 at π/10, back to 0 at π/5 , down to -4 at 3π/10 and then back up to 0 at 2π/5
you did not state in what domain you want these, so I just did the first period.
Of course each maximum happens again in each consecutive or previous period
so our max's happen at x = π/10 + k(2π/5) , where k is an integer
and our min's happen at x = 3π/10 + k(2π/5)
Of course each max will be 4 and each min will be -4
To find the period, amplitude, zeros, and extreme points of each function, we need to identify the key properties of trigonometric functions.
1. Function: y = sin(-2x)
Period: The period of the sine function is given by 2π divided by the absolute value of the coefficient of x in this case, 2. So, the period is 2π/2 = π.
Amplitude: The amplitude of the sine function is always 1. So, the amplitude is 1.
Zeros: Zeros occur when the sine function equals zero. The general formula for zeros is x = kπ, where k is an integer. Since the coefficient of x is negative, this affects the sign of the argument, so the zeros are x = kπ, where k is an integer.
Extreme Points: The maximum and minimum points occur when the sine function reaches its maximum and minimum values, respectively. The maximum value is 1, and it occurs at x = (2k + 1)π/2, where k is an integer. The minimum value is -1, and it occurs at x = kπ, where k is an integer.
2. Function: y = 4sin(5x)
Period: The period is given by 2π divided by the absolute value of the coefficient of x, which is 5. So, the period is 2π/5.
Amplitude: The amplitude is the coefficient in front of the sine function, which is 4. So, the amplitude is 4.
Zeros: Zeros occur when the sine function equals zero. The general formula for zeros is x = kπ, where k is an integer. Since the coefficient of x is positive, this does not affect the sign of the argument, so the zeros are x = kπ, where k is an integer.
Extreme Points: The maximum and minimum values are the amplitude, which is 4 in this case. The maximum value occurs at x = (2k + 1)π/2, where k is an integer, and the minimum value occurs at x = kπ, where k is an integer.
3. Function: y = -3sin(3x/5)
Period: The period is given by 2π divided by the absolute value of the coefficient of x, which is 3/5. So, the period is 2π/(3/5) = (10/3)π.
Amplitude: The amplitude is the coefficient in front of the sine function, which is -3. So, the amplitude is 3.
Zeros: Zeros occur when the sine function equals zero. The general formula for zeros is x = kπ, where k is an integer. Since the coefficient of x is positive, this does not affect the sign of the argument, so the zeros are x = kπ, where k is an integer.
Extreme Points: The maximum and minimum values are the amplitude, which is 3 in this case. The maximum value occurs at x = (2k + 1)π/2, where k is an integer, and the minimum value occurs at x = kπ, where k is an integer.
4. Function: y = 1/2cos(-4x)
Period: The period is given by 2π divided by the absolute value of the coefficient of x, which is 4. So, the period is 2π/4 = π/2.
Amplitude: The amplitude is the coefficient in front of the cosine function, which is 1/2. So, the amplitude is 1/2.
Zeros: Zeros occur when the cosine function equals zero. The general formula for zeros is x = (k + 1/2)π, where k is an integer. Since the coefficient of x is negative and affects the sign of the argument, the zeros are x = (k + 1/2)π, where k is an integer.
Extreme Points: The maximum value occurs at x = kπ, where k is an integer, and the value is equal to the amplitude, which is 1/2. The minimum value occurs at x = (2k + 1)π/2, where k is an integer, and the value is equal to the negative amplitude, which is -1/2.
To find the period, amplitude, zeros, and extreme points at maximum and minimum of the given functions, let's break down each function one by one:
1. y = sin(-2x)
- The general form of a sine function is y = A sin(Bx), where A represents the amplitude and B is related to the period.
- The period of a sine function is given by the formula T = 2π/B, where B is the coefficient of x.
In this case, B = -2. So, the period is T = 2π/(-2) = -π.
However, since the period should always be a positive value, we take the absolute value of -π, giving us π as the period.
- The amplitude of a sine function is the absolute value of the coefficient A. Here, A = 1, so the amplitude is 1.
- Zeros are the x-values where the function intersects the x-axis (y = 0). In the given function, y = sin(-2x), the function will cross the x-axis every half of the period, so the zeros occur at x = π/(-2) and x = 2π/(-2) which simplifies to x = -π/2 and x = -π respectively.
- The maximum and minimum points can be identified by analyzing the amplitude and phase shift:
- Since the amplitude is 1, the maximum point will be y = 1 and the minimum point will be y = -1.
- The function y = sin(-2x) has no phase shift because there is no horizontal shift indicated.
Therefore, the maximum and minimum points occur at x = -π/2, x = -π, and so on, with corresponding y-values of 1 and -1.
2. y = 4sin(5x)
- The coefficient B = 5, so the period T = 2π/5.
- The coefficient A = 4, which indicates an amplitude of 4.
- The zeros occur at x = 0, x = π/5, x = 2π/5, and so on.
- The maximum and minimum points occur at x = π/10, x = 3π/10, etc., with corresponding y-values of 4 and -4.
3. y = -3sin(3x/5)
- The coefficient B = 3/5, so the period T = 2π/(3/5) = 10π/3.
- The coefficient A = -3, indicating an amplitude of 3.
- The zeros occur at x = 0, x = 2π/3, x = 4π/3, and so on.
- The maximum and minimum points occur at x = π/6, x = 7π/6, etc., with corresponding y-values of -3 and 3.
4. y = 1/2cos(-4x)
- The coefficient of x is -4, indicating a period of T = 2π/(-4) = -π/2.
Taking the absolute value, the period is π/2.
- The coefficient A = 1/2, indicating an amplitude of 1/2.
- The zeros occur at x = -π/8, x = 3π/8, etc.
- The maximum and minimum points occur at x = 0, x = π/4, etc., with corresponding y-values of 1/2 and -1/2.
Note: The terms "extreme points" and "zeros" are used interchangeably when referring to the x-values where a function crosses the x-axis.