Can someone please help me explain in words how to what each piece of the equation modifies the whole thing in this:
f(x)=-3csc(2x)+1
huh? I can't even make sens of your words. Hmmm. It appears you are looking for the transformations performed on g(x) = csc(x)
So, just as you have studied with polynomials,
csc(2x) compresses horizontally by a factor of 2
3csc(2x) stretches vertically by a factor of 3
-3csc(2x) reflects over the x-axis
-3csc(2x)+1 translates up by 1
Thank you! Sorry about my bad grammar, I forgot to proof read.
Sure! Let's break down the equation step by step:
1. The function is denoted as f(x), which means it represents the output value of the equation based on the input value of x.
2. The first term in the equation is "-3csc(2x)". csc(2x) is the cosecant function, which is the reciprocal of the sine function. It is calculated as the ratio of the hypotenuse to the opposite side of a right triangle.
3. The factor of "-3" in front of csc(2x) acts as a coefficient that scales the value of csc(2x). Multiplying a value by -3 means it will be reflected across the x-axis and its magnitude will be increased by a factor of 3.
4. The second term in the equation is "+1". This is a constant term that is added to the result of the first term. Its purpose is to shift the entire graph vertically upward by 1 unit. Adding 1 to any value of f(x) will move the graph upward by 1 unit.
By combining these elements, the equation as a whole represents a function f(x) that is the result of scaling the reciprocal of the sine of (2x) by -3, and then shifting the resulting graph upward by 1 unit.