Consider the quadratic function:
[tex]f(x)=x^2-a x-0[/tex]
Vertex: [tex]\left(\frac{-b}{2 a}, f\left(\frac{-b}{2 a}\right)\right)[/tex]
What is the vertex of the function?
Answer (1)
Identify the coefficients of the quadratic function: f ( x ) = x 2 − a x , where the coefficient of x 2 is 1 and the coefficient of x is − a .
Calculate the x-coordinate of the vertex using the formula x v = 2 a − b = 2 a .
Substitute the x-coordinate into the function to find the y-coordinate: y v = f ( 2 a ) = − 4 a 2 .
State the vertex of the function: ( 2 a , − 4 a 2 )
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 − a x − 0 = x 2 − a x . We need to find the vertex of this function. The vertex form of a quadratic function is given by ( 2 a − b , f ( 2 a − b )) , where a and b are the coefficients of the quadratic and linear terms, respectively.
Identifying Coefficients In the given function f ( x ) = x 2 − a x , we can identify the coefficients as follows:
Coefficient of x 2 : a ′ = 1 Coefficient of x : b = − a Constant term: c = 0
Note that we use a ′ to denote the coefficient of x 2 to avoid confusion with the parameter a in the function.
Calculating the x-coordinate of the Vertex The x-coordinate of the vertex is given by the formula: x v = 2 a ′ − b = 2 ( 1 ) − ( − a ) = 2 a So, the x-coordinate of the vertex is 2 a .
Calculating the y-coordinate of the Vertex To find the y-coordinate of the vertex, we substitute the x-coordinate into the function: y v = f ( x v ) = f ( 2 a ) = ( 2 a ) 2 − a ( 2 a ) = 4 a 2 − 2 a 2 = 4 a 2 − 2 a 2 = − 4 a 2 So, the y-coordinate of the vertex is − 4 a 2 .
Final Answer Therefore, the vertex of the quadratic function f ( x ) = x 2 − a x is given by the coordinates: ( 2 a , − 4 a 2 )
Examples
Understanding the vertex of a quadratic function is crucial in various real-world applications. For instance, consider a projectile's trajectory modeled by a quadratic equation. The vertex represents the highest point the projectile reaches. Similarly, in business, if a company's profit is modeled by a quadratic function, the vertex indicates the point of maximum profit. Knowing how to find the vertex allows us to determine these critical points, aiding in decision-making and optimization.
