To analyze the function y = 2x2 + 32x + 4, we first need to identify whether it has a maximum or minimum value by looking at the coefficient of the x2 term. Since the coefficient (2) is positive, this means the parabola opens upwards, indicating that the function has a minimum value.
Next, we use the vertex formula to find the x-coordinate of the vertex, which gives us the minimum value of the function. The x-coordinate of the vertex is calculated using the formula:
x = -b / (2a)
In this case, a = 2 and b = 32.
Plugging in the values, we get:
x = -32 / (2 * 2) = -32 / 4 = -8
Now, we substitute this x-coordinate back into the function to find the minimum value:
y = 2(-8)2 + 32(-8) + 4
y = 2(64) – 256 + 4 = 128 – 256 + 4 = -124
Thus, the minimum value of the function is -124 while the x-coordinate at which this minimum occurs is -8.
For the domain, since it is a quadratic function, it is defined for all real numbers. Therefore, the domain is:
Domain: (-∞, ∞)
For the range, since the minimum value of the function is -124 and the parabola opens upwards, the range starts from -124 and goes to positive infinity. Therefore, the range is:
Range: [-124, ∞)