Consider the quadratic function:
[tex]$f(x)=x^2-8 x-9$[/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: a = 1 , b = − 8 , and c = − 9 .
Calculate the x-coordinate of the vertex: x v = 2 a − b = 2 ( 1 ) − ( − 8 ) = 4 .
Calculate the y-coordinate of the vertex: y v = f ( 4 ) = ( 4 ) 2 − 8 ( 4 ) − 9 = − 25 .
State the vertex: The vertex of the function is ( 4 , − 25 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 − 8 x − 9 . We need to find the vertex of this function. The vertex form is given by ( 2 a − b , f ( 2 a − b ) ) , where a , b , and c are the coefficients of the quadratic function in the form f ( x ) = a x 2 + b x + c . In our case, a = 1 , b = − 8 , and c = − 9 .
Calculating the x-coordinate First, we calculate the x-coordinate of the vertex using the formula x v = 2 a − b . Substituting the values a = 1 and b = − 8 , we get: x v = 2 ( 1 ) − ( − 8 ) = 2 8 = 4
Calculating the y-coordinate Next, we calculate the y-coordinate of the vertex by evaluating the function f ( x ) at x = x v = 4 . y v = f ( 4 ) = ( 4 ) 2 − 8 ( 4 ) − 9 = 16 − 32 − 9 = − 16 − 9 = − 25
Final Answer Therefore, the vertex of the quadratic function is ( 4 , − 25 ) .
Examples
Understanding the vertex of a quadratic function is useful in various real-world applications. For example, if the function represents the height of a projectile over time, the vertex gives the maximum height reached by the projectile and the time at which it reaches that height. Similarly, in business, if a quadratic function represents the profit of a company as a function of the number of units sold, the vertex gives the number of units that maximize the profit and the maximum profit itself. This helps in optimizing processes and making informed decisions.
