What is the vertex of the function [tex]f(x)=\frac{1}{2} x^2+3 x+\frac{3}{2}[/tex]?
Answer (1)
Identify the coefficients: a = 2 1 , b = 3 , and c = 2 3 .
Calculate the x-coordinate of the vertex: x v = − 2 a b = − 3 .
Calculate the y-coordinate of the vertex: y v = f ( − 3 ) = − 3 .
State the vertex: ( − 3 , − 3 ) .
Explanation
Problem Analysis We are given the quadratic function f ( x ) = 2 1 x 2 + 3 x + 2 3 . Our goal is to find the vertex of this function.
Finding the x-coordinate of the Vertex The general form of a quadratic function is f ( x ) = a x 2 + b x + c . In our case, we have a = 2 1 , b = 3 , and c = 2 3 . The x-coordinate of the vertex, denoted as x v , can be found using the formula x v = − 2 a b .
Calculating the x-coordinate Substituting the values of a and b into the formula, we get: x v = − 2 × 2 1 3 = − 1 3 = − 3.
Finding the y-coordinate of the Vertex Now, we need to find the y-coordinate of the vertex, denoted as y v . We can do this by substituting the value of x v into the original function f ( x ) :
y v = f ( x v ) = f ( − 3 ) = 2 1 ( − 3 ) 2 + 3 ( − 3 ) + 2 3 .
Calculating the y-coordinate Let's calculate y v :
y v = 2 1 ( 9 ) − 9 + 2 3 = 2 9 − 2 18 + 2 3 = 2 9 − 18 + 3 = 2 − 6 = − 3.
Final Answer Therefore, the vertex of the function f ( x ) = 2 1 x 2 + 3 x + 2 3 is ( − 3 , − 3 ) .
Examples
Understanding the vertex of a quadratic function is very useful in real-world applications. For example, if you are launching a projectile, the vertex of the parabolic trajectory represents the maximum height the projectile will reach. Similarly, in business, if a cost function is quadratic, the vertex can represent the point of minimum cost. Knowing how to find the vertex allows you to optimize various scenarios, whether it's maximizing height or minimizing costs.
