1. To obtain the graph of y = asinb (x - h) + k ,
explain what each of the values of a, b, h, and k affects the graph of function y = sin x
If you go back to y = x^2 the quadratic you know that it opens upwards and has a vertex at (0,0), you also know if you take y=x^2 +7 this is a vertical translation up 7 units.
So my question to you is where in your function is the translation vertically located?
To understand how the values of a, b, h, and k affect the graph of the function y = asinb(x - h) + k, we need to understand the general form of the function and how each parameter modifies it.
1. The value of "a" affects the amplitude of the graph. It amplifies or shrinks the height of the sinusoidal function. When a > 1, the graph becomes stretched vertically, while when 0 < a < 1, the graph becomes compressed vertically. If a is negative (-a), the graph will be reflected about the x-axis.
2. The value of "b" changes the period of the graph. The period is the length it takes for one complete cycle of the sine curve. The period of the function y = sin x is 2π. When b > 1, the graph compresses horizontally, making the curve complete one cycle in less than 2π. Conversely, when 0 < b < 1, the graph stretches horizontally, making the curve complete one cycle in more than 2π. If b is negative (-b), the graph will be reflected about the y-axis.
3. The value of "h" represents a horizontal shift or phase shift of the graph. It moves the entire graph left or right based on the value of h. If h > 0, the graph shifts to the right, and if h < 0, the graph shifts to the left.
4. The value of "k" represents a vertical shift of the graph. It moves the graph up or down based on the value of k. If k > 0, the graph shifts upwards, and if k < 0, the graph shifts downwards.
When combined, all these parameters modify the graph of the function y = asinb(x - h) + k in various ways, altering the amplitude, the period, and the position of the curve.