For the graph, what is a reasonable constraint so that the function is at least 300?
graph of y equals minus 2 times the square of x plus 50 times x plus 300
x ≥ 0
−5 ≤ x ≤ 30
0 ≤ x ≤ 25
All real numbers
0 ≤ x ≤ 25
To find a reasonable constraint so that the function is at least 300, we need to determine the range of x-values for which the y-value is greater than or equal to 300.
Looking at the equation, y = -2x^2 + 50x + 300, we can determine the vertex of the parabola to find the minimum y-value.
The x-coordinate of the vertex is given by: x = -b/2a
where a = -2 and b = 50.
x = -(50) / 2(-2)
x = -50 / -4
x = 12.5
Therefore, the vertex occurs at x = 12.5.
To find the y-coordinate of the vertex, substitute this into the equation: y = -2(12.5)^2 + 50(12.5) + 300.
y = -2(156.25) + 625 + 300
y = -312.5 + 625 + 300
y = 612.5
Therefore, the minimum y-value is 612.5.
To ensure that the function is at least 300, we need to find the range of x-values for which the y-value is greater than or equal to 300.
Since the parabola opens downwards, we know that for x-values outside the range of the vertex, the y-values will be less than 612.5. Therefore, we need to find the range of x-values for which the y-values are greater than or equal to 300.
Looking at the graph, it appears that the range of x-values that satisfy this constraint is approximately 0 ≤ x ≤ 20.
Therefore, a reasonable constraint so that the function is at least 300 is 0 ≤ x ≤ 20.