The range of y=asinx+b is {y| -1 ≤ y ≤ 4, yeR}.
If a is positive, then the value of b is:
a) 3/2
b) -1
c) 4
d) 5/2
How do I solve this question?
max value = a + b
min value = -a + b
so
b-a = 1
b+a = 4
---------
2 b = 5
To solve this question, we need to understand the properties of the sine function and how it affects the range of a function.
The sine function, sin(x), oscillates between -1 and 1. So, without any transformations, the range of the function y = sin(x) would be between -1 and 1.
In the given equation y = asin(x) + b, the constant term "b" shifts the entire graph vertically, but it does not affect the oscillation or amplitude of the function. The amplitude of the sine function remains as "a".
Given that the range of y is -1 ≤ y ≤ 4, it means that the minimum value of y is -1 and the maximum value of y is 4.
Now, let's consider two extreme cases to determine the range:
1. When sin(x) = -1, the lowest value of y would be:
y = asin(x) + b
-1 = a(-1) + b
-1 = -a + b
2. When sin(x) = 1, the highest value of y would be:
y = asin(x) + b
1 = a(1) + b
1 = a + b
From the above two equations, we can conclude that "a + b" gives us the maximum value of y, and "-a + b" gives us the minimum value of y.
Based on the range given (-1 ≤ y ≤ 4), we can write these two inequalities:
-1 ≤ -a + b ----- (1)
4 ≥ a + b ----- (2)
Now, since "a" is positive, we can multiply equation (1) by -1 to change the direction of the inequality sign:
1 ≥ a - b ----- (3)
By examining equations (2) and (3), we can see that the value of "b" must be within the range [1, 4].
Out of the answer options provided, the only value of "b" within this range is 4. Therefore, the correct answer is c) 4.