To sketch a graph of the rate against y, we can analyze the factors in the equation: rate = k(a-y)(b-y).
First, let's consider the factors (a-y) and (b-y). From the given information, we know that a > y and b > y.
When y is less than both a and b, both factors (a-y) and (b-y) are positive. As y increases, the factors decrease until y reaches either a or b. At y = a, the factor (a-y) becomes zero, and at y = b, the factor (b-y) becomes zero.
Now, consider the factor (a-y)(b-y). When both factors are positive (y < a and y < b), the product is positive. When one factor becomes zero (y = a or y = b) while the other factor is still positive, the product remains zero. When both factors become zero at y = a and y = b, the product remains zero.
Based on this analysis, we can conclude the following:
1. When y < a and y < b, the rate is nonnegative (positive or zero).
2. When y = a or y = b, the rate is zero.
3. When y > a or y > b, the rate is negative.
To summarize, the rate is nonnegative for values of y that are less than both a and b, i.e., y < a and y < b.