Consider the following function:
[tex]q(x)=-x^2+6 x-8[/tex]
Step 1 of 4: Find the vertex.
Answer (2)
Find the x-coordinate of the vertex using the formula x = − 2 a b , where a = − 1 and b = 6 , which gives x = 3 .
Substitute x = 3 into the function q ( x ) = − x 2 + 6 x − 8 to find the y-coordinate of the vertex: q ( 3 ) = − ( 3 ) 2 + 6 ( 3 ) − 8 = 1 .
The vertex of the quadratic function is ( 3 , 1 ) .
Therefore, the vertex is ( 3 , 1 ) .
Explanation
Understanding the Problem We are given the quadratic function q ( x ) = − x 2 + 6 x − 8 and asked to find its vertex. The vertex of a quadratic function in the form a x 2 + b x + c is the point where the function reaches its maximum or minimum value. The x-coordinate of the vertex can be found using the formula x = − 2 a b .
Finding the x-coordinate In our case, a = − 1 and b = 6 . Plugging these values into the formula, we get: x = − 2 ( − 1 ) 6 = − − 2 6 = 3.
Finding the y-coordinate Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting x = 3 into the function q ( x ) :
q ( 3 ) = − ( 3 ) 2 + 6 ( 3 ) − 8 = − 9 + 18 − 8 = 1.
Stating the Vertex Therefore, the vertex of the quadratic function q ( x ) = − x 2 + 6 x − 8 is ( 3 , 1 ) .
Examples
Understanding quadratic functions and their vertices is crucial in various real-world applications. For instance, if you're launching a projectile, the quadratic function can model its trajectory, and the vertex represents the maximum height the projectile reaches. Similarly, in business, if a quadratic function describes the profit of a product, the vertex indicates the production level that maximizes profit. Knowing how to find the vertex allows you to optimize outcomes in these scenarios.
The vertex of the quadratic function q ( x ) = − x 2 + 6 x − 8 is found by calculating the x-coordinate using the formula x = − 2 a b , resulting in x = 3 . By substituting this value back into the function, the y-coordinate is determined to be 1. Hence, the vertex is ( 3 , 1 ) .
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