What is the vertex of the quadratic function $f(x)=(x-6)(x+2)$?
Answer (2)
Expand the quadratic function: f ( x ) = ( x − 6 ) ( x + 2 ) = x 2 − 4 x − 12 .
Find the x-coordinate of the vertex: x v = − 2 a b = − 2 ( 1 ) − 4 = 2 .
Find the y-coordinate of the vertex: y v = f ( 2 ) = ( 2 ) 2 − 4 ( 2 ) − 12 = − 16 .
The vertex of the quadratic function is ( 2 , − 16 ) .
Explanation
Expanding the Quadratic Function We are given the quadratic function f ( x ) = ( x − 6 ) ( x + 2 ) and we want to find its vertex. The vertex of a quadratic function is the point where the function reaches its minimum or maximum value. To find the vertex, we first need to expand the quadratic function into the standard form f ( x ) = a x 2 + b x + c .
Expanded Form Expanding the given function, we have:
f ( x ) = ( x − 6 ) ( x + 2 ) = x ( x + 2 ) − 6 ( x + 2 ) = x 2 + 2 x − 6 x − 12 = x 2 − 4 x − 12 .
So, f ( x ) = x 2 − 4 x − 12 .
Finding the x-coordinate of the Vertex Now that we have the quadratic function in the form f ( x ) = a x 2 + b x + c , we can identify the coefficients: a = 1 , b = − 4 , and c = − 12 . The x-coordinate of the vertex, x v , can be found using the formula x v = − 2 a b .
Calculating x_v Substituting the values of a and b into the formula, we get:
x v = − 2 ( 1 ) − 4 = 2 4 = 2 .
Finding the y-coordinate of the Vertex Now that we have the x-coordinate of the vertex, we can find the y-coordinate, y v , by substituting x v back into the function f ( x ) .
y v = f ( x v ) = f ( 2 ) = ( 2 ) 2 − 4 ( 2 ) − 12 = 4 − 8 − 12 = − 16 .
The Vertex Therefore, the vertex of the quadratic function is ( 2 , − 16 ) .
Examples
Understanding the vertex of a quadratic function is useful in many real-world applications. For example, if you are 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 various processes and make informed decisions.
The vertex of the quadratic function f ( x ) = ( x − 6 ) ( x + 2 ) is found by expanding it to f ( x ) = x 2 − 4 x − 12 and using the vertex formula, yielding the vertex at ( 2 , − 16 ) .
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