What is the maximum or minimum value of the function? What is the range?
y = –2x2 + 32x –12
(1 point)
Responses
maximum: 116
range: y A symbol less than or equal to116
maximum: 116 range: y Image with alt text: A symbol less than or equal to 116
maximum: –116
range: y greater than equal to–116
maximum: –116 range: y Image with alt text: greater than equal to –116
maximum: 116
range: y greater than equal to116
maximum: 116 range: y Image with alt text: greater than equal to 116
maximum: –116
range: y A symbol less than or equal to–116
maximum: 116
range: y greater than or equal to -116
To find the maximum or minimum value of the function y = -2x^2 + 32x - 12, we need to determine whether the parabola opens upwards or downwards.
The coefficient of the x^2 term is -2, which means the parabola opens downwards. In this case, the maximum value will occur at the vertex of the parabola.
To find the x-coordinate of the vertex, we can use the formula x = -b / (2a), where a and b are the coefficients of the x^2 and x terms respectively.
In this case, a = -2 and b = 32, so x = -32 / (2 * (-2)) = -32 / (-4) = 8.
Now, substitute this value of x back into the equation y = -2x^2 + 32x - 12 to find the y-coordinate of the vertex:
y = -2(8)^2 + 32(8) - 12 = -2(64) + 256 - 12 = -128 + 256 - 12 = 116.
Therefore, the maximum value of the function is 116.
To find the range, we consider the y-values that the function can take. Since the parabola opens downwards, the range will be from negative infinity up to the maximum value of 116, inclusive. So the range of the function is y ≤ 116.
To find the maximum or minimum value of the function and the range, we need to determine the vertex of the quadratic equation represented by the function.
The vertex of a quadratic equation in the form of y = ax^2 + bx + c can be found using the formula x = -b / (2a). Once we find the x-coordinate of the vertex, we can substitute it back into the equation to find the corresponding y-coordinate.
In the given function y = -2x^2 + 32x - 12, we can identify that a = -2, b = 32, and c = -12.
Using the formula x = -b / (2a) = -32 / (2 * -2) = 8, we determine the x-coordinate of the vertex is 8.
Substituting x = 8 into the equation, we find y = -2(8)^2 + 32(8) - 12 = -128 + 256 - 12 = 116. Therefore, the y-coordinate of the vertex is 116.
So, the maximum or minimum value of the function is 116 (maximum, in this case).
To find the range, we consider the graph of the quadratic equation. Since the coefficient of the x^2 term (-2) is negative, the parabola opens downward and the vertex represents the maximum point. Therefore, the range of the function is all values less than or equal to 116.
Hence, the correct answer is:
maximum: 116
range: y ≤ 116