Write a rule for a nonlinear function such that y is negative when x = 1, positive when x = 2, negative when x = 3, positive when x = 4, and so on.
The entire problem confuses me. I have no idea what its even asking besides it wants an equation.
Would it be y=(-1)^x?
You got it!
its 2022
To create a rule for a nonlinear function that satisfies the given conditions, we can use a trigonometric function. Let's choose the sine function.
The equation for the sine function is: y = A * sin(B * x + C) + D, where A, B, C, and D are constants.
To meet the given conditions, we need to specify the values of A, B, C, and D. Let's start by analyzing the given conditions:
- y is negative when x = 1: This means that y should be negative when we substitute 1 into the equation.
- y is positive when x = 2: This means that y should be positive when we substitute 2 into the equation.
- y is negative when x = 3: This means that y should be negative when we substitute 3 into the equation.
- y is positive when x = 4: This means that y should be positive when we substitute 4 into the equation.
- and so on...
Based on these conditions, we can make the following observations:
- When x = 1, B * x + C should be π or any odd multiple of π (i.e., π, 3π, 5π, etc.) in order for the sine function to be negative.
- When x = 2, B * x + C should be 0 or any even multiple of π (i.e., 0, 2π, 4π, etc.) in order for the sine function to be positive.
- When x = 3, B * x + C should be π or any odd multiple of π (i.e., π, 3π, 5π, etc.) in order for the sine function to be negative.
- When x = 4, B * x + C should be 0 or any even multiple of π (i.e., 0, 2π, 4π, etc.) in order for the sine function to be positive.
To satisfy these observations, we can choose B = π/2 (or any odd multiple of π/2) and C = 0.
Now, let's consider the A and D values. Since we want the function to have alternating positive and negative values, we can set A = 1 and D = 0.
Putting it all together, our rule for the nonlinear function becomes:
y = sin((π/2) * x)
This equation will generate a graph where y is negative at x = 1, positive at x = 2, negative at x = 3, positive at x = 4, and so on, as required.