Answer:
y = 4 sin(2x + π/4) + 3
Explanation:
A sinusoidal function is a function that has the form y = a sin(bx + c) + d or y = a cos(bx + c) + d, where a, b, c, and d are constants. The range of a sinusoidal function is the set of possible values of y for any value of x. The range depends on the values of a and d, which are the amplitude and the vertical shift of the function respectively.
To write the equation of a sinusoidal function with a range of [-1, 7], we need to find the values of a and d that satisfy the following inequalities:
- -1 ≤ a sin(bx + c) + d ≤ 7
- -1 - d ≤ a sin(bx + c) ≤ 7 - d
One possible way to find a and d is to use the fact that the range of sin(bx + c) is [-1, 1] for any values of b and c. Therefore, we can multiply both sides of the inequality by a positive number to get the desired range. For example, if we multiply by 4, we get:
Then, we can add a constant to both sides of the inequality to shift the range up or down. For example, if we add 3, we get:
- -1 ≤ 4 sin(bx + c) + 3 ≤ 7
This satisfies the given range, so we can choose a = 4 and d = 3. We can also choose any values for b and c, as long as b is not zero. For example, if we choose b = 2 and c = π/4, we get:
This is one possible equation of a sinusoidal function with a range of [-1, 7]. There are infinitely many other equations that have the same range, but different values of a, b, c, and d. We can also use cosine instead of sine to write another equation with the same range. For example: