What is the range of the function $f(x)=3 x^2+6 x-8$ ?
A. {y | y ≥ -1}
B. {y | y ≤ -1}
C. {y | y ≥ -11}
D. {y | y ≤ -11}
Answer (1)
Find the x-coordinate of the vertex: x v = − 2 a b = − 1 .
Find the y-coordinate of the vertex: y v = f ( − 1 ) = − 11 .
Since the parabola opens upwards, the range is all y ≥ − 11 .
The range of the function is y ∣ y ≥ − 11 .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 3 x 2 + 6 x − 8 and asked to find its range. The range of a quadratic function is the set of all possible output values (y-values). Since the coefficient of the x 2 term is positive ( 0"> 3 > 0 ), the parabola opens upwards, meaning it has a minimum value. The minimum value occurs at the vertex of the parabola.
Finding the Vertex x-coordinate To find the vertex, we first find the x-coordinate of the vertex using the formula x v = − 2 a b , where a = 3 and b = 6 . Thus, x v = − 2 ( 3 ) 6 = − 6 6 = − 1.
Finding the Vertex y-coordinate Next, we find the y-coordinate of the vertex by plugging x v = − 1 into the function: y v = f ( − 1 ) = 3 ( − 1 ) 2 + 6 ( − 1 ) − 8 = 3 ( 1 ) − 6 − 8 = 3 − 6 − 8 = − 11. So, the vertex of the parabola is ( − 1 , − 11 ) .
Determining the Range Since the parabola opens upwards, the minimum value of the function is − 11 . Therefore, the range of the function is all y-values greater than or equal to − 11 . In set notation, this is written as y ∣ y ≥ − 11 .
Examples
Imagine you're designing a parabolic arch for a bridge. The function f ( x ) = 3 x 2 + 6 x − 8 models the height of the arch at different points. Finding the range of this function tells you the minimum height of the arch, which is crucial for ensuring there's enough clearance underneath. Understanding the range helps engineers design safe and functional structures.
