$f(x)=x^2-8 x-9$
Vertex: $\left(\frac{-b}{2 a}, f\left(\frac{-b}{2 a}\right)\right)$
Answer (1)
Identify the coefficients: a = 1 , b = − 8 , 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 quadratic function is ( 4 , − 25 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 − 8 x − 9 and the formula for the vertex of a parabola ( 2 a − b , f ( 2 a − b ) ) . Our goal is to find the vertex of this quadratic function.
Identifying Coefficients First, we need to identify the coefficients a , b , and c in the quadratic function f ( x ) = a x 2 + b x + c . In this case, we have a = 1 , b = − 8 , and c = − 9 .
Calculating the x-coordinate Next, we calculate the x-coordinate of the vertex using the formula x v = 2 a − b . Substituting the values of a and b , we get x v = 2 ( 1 ) − ( − 8 ) = 2 8 = 4.
Calculating the y-coordinate Now, we substitute the value of x v into the function f ( x ) to find the y-coordinate of the vertex, y v = f ( x v ) .
y v = f ( 4 ) = ( 4 ) 2 − 8 ( 4 ) − 9 = 16 − 32 − 9 = − 16 − 9 = − 25.
Stating the Vertex Therefore, the vertex of the quadratic function is ( 4 , − 25 ) .
Examples
Understanding the vertex of a parabola is crucial in various real-world applications. For instance, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if you have a cost function that is quadratic, the vertex can represent the point at which you minimize your costs or maximize your profits. Knowing how to find the vertex allows you to optimize these kinds of scenarios.
