Find the vertex of the quadratic function: [tex]f(x)=-3 x^2+12 x+1[/tex]
Answer (1)
Identify the coefficients: a = − 3 , b = 12 , and c = 1 .
Calculate the x-coordinate of the vertex: x = − 2 a b = 2 .
Substitute x = 2 into the function to find the y-coordinate: f ( 2 ) = 13 .
State the vertex: ( 2 , 13 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = − 3 x 2 + 12 x + 1 , and we want to find its vertex. The vertex of a quadratic function in the form f ( x ) = a x 2 + b x + c is a point ( x , y ) where x = − 2 a b and y = f ( x ) .
Finding the x-coordinate In our case, a = − 3 , b = 12 , and c = 1 . Let's find the x-coordinate of the vertex: x = − 2 a b = − 2 ( − 3 ) 12 = − − 6 12 = 2
Finding the y-coordinate Now that we have the x-coordinate, we can find the y-coordinate by plugging x = 2 into the function: f ( 2 ) = − 3 ( 2 ) 2 + 12 ( 2 ) + 1 = − 3 ( 4 ) + 24 + 1 = − 12 + 24 + 1 = 13
Stating the Vertex Therefore, the vertex of the quadratic function is ( 2 , 13 ) .
Examples
Understanding quadratic functions and their vertices is crucial in various real-world applications. For instance, if you're launching a projectile, the vertex of the projectile's parabolic path represents the maximum height it will reach. Similarly, in business, if you model profit as a quadratic function of production quantity, the vertex will indicate the production level that maximizes profit. Knowing how to find the vertex allows you to optimize outcomes in these scenarios.
