What is the maximum or minimum value of the function? What is the range?
y = –2x2 + 32x –12 (1 point)
To find the maximum or minimum value of the function, we can use the vertex formula. The vertex formula is given by x = -b/2a, where a, b, and c are the coefficients of the function.
In the given function, a = -2 and b = 32. Substituting these values into the vertex formula, we have x = -32/(2*(-2)) = -32/(-4) = 8.
To find the y-coordinate of the vertex, we substitute the value of x back into the function: y = -2(8)^2 + 32(8) - 12 = -128 + 256 - 12 = 116.
Therefore, the vertex of the function is (8, 116). Since the coefficient of x^2 is negative, the parabola opens downwards, meaning the vertex represents the maximum point of the function.
Hence, the maximum value of the function is 116.
To determine the range of the function, we consider the behavior of the parabola. Since the parabola opens downwards, the range will be all y-values less than or equal to the maximum value.
Therefore, the range of the function is y ≤ 116.
To find the maximum or minimum value of a quadratic function in the form of y = ax^2 + bx + c, we need to determine the vertex of the parabola. The vertex represents the highest or lowest point on the graph, depending on whether the parabola opens upward or downward.
The equation of the given quadratic function is y = -2x^2 + 32x - 12. We can use the formula -b/2a to find the x-coordinate of the vertex.
For this equation, a = -2 and b = 32.
The x-coordinate of the vertex is given by x = -b / (2a) = -32 / (2*(-2)) = -32 / (-4) = 8.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the equation:
y = -2(8)^2 + 32(8) - 12 = -2(64) + 256 - 12 = -128 + 256 - 12 = 116.
Therefore, the vertex of the function is (8, 116).
Since the parabola opens downward (leading coefficient is negative), the vertex represents the maximum value of the function.
The maximum value of the function is y = 116.
To find the range of the function, note that since the parabola opens downward, the range is from negative infinity up to the maximum value. Thus, the range of the function is (-∞, 116].
To find the maximum or minimum value of the function, we need to determine the vertex of the parabola. The vertex of a quadratic function in the form of y = ax^2 + bx + c can be found using the formula:
x = -b/2a
For the given function, y = -2x^2 + 32x - 12, a = -2, b = 32, and c = -12.
x = -32 / (2*-2)
x = -32 / -4
x = 8
To find the value of y at x = 8, we substitute the value of x back into the equation:
y = -2(8)^2 + 32(8) - 12
y = -2(64) + 256 - 12
y = -128 + 256 - 12
y = 116
Therefore, the maximum or minimum value of the function is 116.
To find the range, we consider the shape of the quadratic function. Since the coefficient of x^2 is negative, the parabola opens downwards, and the range is from negative infinity up to the maximum value. So, the range of the function is (-∞, 116].