To find the range of the quadratic function \( y = x^2 + 12x + 20 \), we can start by rewriting the quadratic in vertex form. This will allow us to easily identify the vertex, which gives us the minimum (or maximum) value of the function.
-
Completing the square:
-
Start with the original function: \[ y = x^2 + 12x + 20 \]
-
Rearranging the first two terms: \[ y = (x^2 + 12x) + 20 \]
-
To complete the square, take half of the coefficient of \( x \) (which is 12), square it and then add it inside the parentheses: \[ \left(\frac{12}{2}\right)^2 = 36 \]
-
Add and subtract 36 inside the parentheses: \[ y = (x^2 + 12x + 36 - 36) + 20 \] \[ y = (x + 6)^2 - 36 + 20 \] \[ y = (x + 6)^2 - 16 \]
-
-
Identifying the vertex:
The quadratic is now in vertex form \( y = (x + 6)^2 - 16 \), where the vertex is at the point \( (-6, -16) \).
-
Determining the range:
Since the coefficient of \( (x + 6)^2 \) is positive, the parabola opens upwards. Thus, the minimum value of \( y \) occurs at the vertex. Therefore, the minimum value of \( y \) is \( -16 \).
Given that the parabola opens upwards, the range of the function starts from this minimum value:
\[ \text{Range: } y \geq -16 \]
In interval notation, the range is: \[ \text{Range: } [-16, \infty) \]
Thus, the range of the quadratic function \( y = x^2 + 12x + 20 \) is \([-16, \infty)\).