how do i find out what the amplitude, period, h, k and midline of problems
1) y= sin (x-5)=7
2) y=3 cos (x- pie) -1
3) y= Sin3x+4
4) y= 1/2 cos 2x-2
To find the amplitude, period, h, k, and midline of a function, you need to understand their definitions and then analyze the given equations step by step. Here’s how you can find these values for each problem:
1) y = sin(x - 5) + 7:
Amplitude (A): The amplitude represents the maximum distance the graph reaches above or below the midline. For a sine function, the amplitude is always positive and equal to the absolute value of the coefficient in front of the trigonometric function. In this case, the amplitude is 1.
Period (T): The period is the amount of time it takes the graph to complete one full cycle. For a sine function, the period can be found using the formula 2π/|B|, where B is the coefficient of "x" inside the trigonometric function. Here, the coefficient of "x" is 1, so the period is 2π.
h: The value inside the parentheses of the trigonometric function represents a horizontal shift left or right. In this case, h = 5, indicating a shift to the right by 5 units.
k: The value outside the parentheses of the trigonometric function (added at the end) represents a vertical shift up or down. In this case, k = 7, indicating a shift upward by 7 units.
Midline: The midline of a sine function is given by the formula y = k, where k represents the vertical shift. In this case, the midline is y = 7.
2) y = 3 cos(x - π) - 1:
For this equation, the steps are the same as before.
Amplitude (A): The amplitude is equal to the absolute value of the coefficient in front of the trigonometric function. Here, the amplitude is 3.
Period (T): The period can be found using the formula 2π/|B|, where B is the coefficient of "x" inside the trigonometric function. In this case, the coefficient of "x" is 1, so the period is 2π.
h: The value inside the parentheses of the trigonometric function represents a horizontal shift left or right. In this case, h = π, indicating a shift to the right by π units.
k: The value outside the parentheses of the trigonometric function (added at the end) represents a vertical shift up or down. In this case, k = -1, indicating a shift downward by 1 unit.
Midline: The midline is given by the formula y = k, where k represents the vertical shift. In this case, the midline is y = -1.
3) y = sin(3x) + 4:
Amplitude (A): The amplitude is equal to the absolute value of the coefficient in front of the trigonometric function. Here, the amplitude is 1.
Period (T): The period can be found using the formula 2π/|B|, where B is the coefficient of "x" inside the trigonometric function. In this case, the coefficient of "x" is 3, so the period is 2π/3.
h: There is no horizontal shift in this equation, so h = 0.
k: The value outside the parentheses of the trigonometric function (added at the end) represents a vertical shift up or down. In this case, k = 4, indicating a shift upward by 4 units.
Midline: The midline is given by the formula y = k, where k represents the vertical shift. In this case, the midline is y = 4.
4) y = (1/2) cos(2x) - 2:
Amplitude (A): The amplitude is equal to the absolute value of the coefficient in front of the trigonometric function. Here, the amplitude is 1/2.
Period (T): The period can be found using the formula 2π/|B|, where B is the coefficient of "x" inside the trigonometric function. In this case, the coefficient of "x" is 2, so the period is π.
h: There is no horizontal shift in this equation, so h = 0.
k: The value outside the parentheses of the trigonometric function (added at the end) represents a vertical shift up or down. In this case, k = -2, indicating a shift downward by 2 units.
Midline: The midline is given by the formula y = k, where k represents the vertical shift. In this case, the midline is y = -2.
By following these steps, you can determine the amplitude, period, h, k, and midline for each given problem.