Question
What is the range of -x2+x+4.75?
Answers
GPT 3.5
The given expression is a quadratic function, so its range depends on the value of the coefficient of the quadratic term (-x^2).
Since the coefficient of the quadratic term is negative, the parabola opens downwards. The vertex of the parabola is at the maximum point, so the maximum value of the function can be found at the vertex.
To find the vertex of the parabola, we can use the formula for the x-coordinate of the vertex: x = -b/(2a), where a is the coefficient of the quadratic term and b is the coefficient of the linear term.
In this case, a = -1 and b = 1, so x = -1/(2*(-1)) = 1/2. Plugging this value back into the function gives us:
f(1/2) = -(1/2)^2 + (1/2) + 4.75 = -1/4 + 1/2 + 4.75 = 2.5
Therefore, the maximum value of the function is 2.5, which occurs at x = 1/2.
Since the parabola opens downwards, the range of the function is (-∞, 2.5].
Since the coefficient of the quadratic term is negative, the parabola opens downwards. The vertex of the parabola is at the maximum point, so the maximum value of the function can be found at the vertex.
To find the vertex of the parabola, we can use the formula for the x-coordinate of the vertex: x = -b/(2a), where a is the coefficient of the quadratic term and b is the coefficient of the linear term.
In this case, a = -1 and b = 1, so x = -1/(2*(-1)) = 1/2. Plugging this value back into the function gives us:
f(1/2) = -(1/2)^2 + (1/2) + 4.75 = -1/4 + 1/2 + 4.75 = 2.5
Therefore, the maximum value of the function is 2.5, which occurs at x = 1/2.
Since the parabola opens downwards, the range of the function is (-∞, 2.5].