Determine the vertex of the function [tex]f(x)=3 x^2-6 x+13[/tex].
1. Identify the values of [tex]a[/tex] and [tex]b[/tex].
[tex]a=[/tex] $\square$ and [tex]b=[/tex] $\square$
2. Find the [tex]x[/tex]-coordinate of the vertex.
[tex]-\frac{b}{2 a}=[/tex] $\square$
3. Find the [tex]y[/tex]-coordinate by evaluating the function at the [tex]x[/tex]-value found in the previous step.
The vertex is $\square$
Answer (1)
Identify the coefficients a and b from the quadratic function: a = 3 and b = − 6 .
Calculate the x -coordinate of the vertex using the formula x = − 2 a b , which gives x = 1 .
Evaluate the function at x = 1 to find the y -coordinate of the vertex: f ( 1 ) = 10 .
State the vertex as a coordinate point: ( 1 , 10 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 3 x 2 − 6 x + 13 . Our goal is to find the vertex of this parabola. The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. We can find the vertex by using the formula x = − 2 a b to find the x -coordinate of the vertex, and then plugging that value into the function to find the y -coordinate.
Identifying a and b First, we need to identify the values of a and b from the given quadratic function f ( x ) = 3 x 2 − 6 x + 13 . Comparing this to the standard form f ( x ) = a x 2 + b x + c , we can see that a = 3 and b = − 6 .
Finding the x-coordinate Next, we find the x -coordinate of the vertex using the formula x = − 2 a b . Substituting the values of a and b , we get: x = − 2 ( 3 ) − 6 = − 6 − 6 = 1
Finding the y-coordinate Now, we find the y -coordinate of the vertex by evaluating the function f ( x ) at x = 1 : f ( 1 ) = 3 ( 1 ) 2 − 6 ( 1 ) + 13 = 3 − 6 + 13 = 10
Stating the Vertex Therefore, the vertex of the function f ( x ) = 3 x 2 − 6 x + 13 is ( 1 , 10 ) .
Examples
Understanding the vertex of a quadratic function 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 represented by a quadratic equation, the vertex can help you determine the minimum cost. Knowing how to find the vertex allows you to optimize processes and make informed decisions in fields ranging from physics to economics.
