Find the vertex of the parabola.
[tex]f(x)=x^2-6 x+13[/tex]
A. (4,0)
B. (0,3)
C. (3,4)
D. (4,3)
Answer (2)
The vertex of the parabola given by the function f ( x ) = x 2 − 6 x + 13 is found to be ( 3 , 4 ) using both the method of completing the square and the vertex formula. The correct answer is option C (3, 4).
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Rewrite the quadratic function in vertex form by completing the square: f ( x ) = ( x − 3 ) 2 + 4 .
Identify the vertex from the vertex form as ( 3 , 4 ) .
Alternatively, use the formula x = − 2 a b to find the x-coordinate of the vertex, which is x = 3 .
Calculate the y-coordinate by plugging x = 3 into the function: f ( 3 ) = 4 . Thus, the vertex is ( 3 , 4 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 − 6 x + 13 and asked to find the vertex of the parabola it represents. The vertex form of a parabola is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. We can find the vertex by completing the square or 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.
Completing the Square Let's complete the square to rewrite the given quadratic function in vertex form. We have f ( x ) = x 2 − 6 x + 13 . First, we rewrite the function as f ( x ) = ( x 2 − 6 x ) + 13 . To complete the square, we take half of the coefficient of the x term, which is -6/2 = -3, and square it: ( − 3 ) 2 = 9 . We add and subtract 9 inside the parenthesis: f ( x ) = ( x 2 − 6 x + 9 − 9 ) + 13 .
Rewriting in Vertex Form Now, we rewrite the expression as f ( x ) = ( x 2 − 6 x + 9 ) − 9 + 13 . We can factor the quadratic expression: f ( x ) = ( x − 3 ) 2 − 9 + 13 . Simplifying the expression, we get f ( x ) = ( x − 3 ) 2 + 4 .
Identifying the Vertex From the vertex form f ( x ) = ( x − 3 ) 2 + 4 , we can identify the vertex as ( h , k ) = ( 3 , 4 ) . Therefore, the vertex of the parabola is ( 3 , 4 ) .
Using the Vertex Formula Alternatively, we can use the formula x = − 2 a b to find the x-coordinate of the vertex. In the given quadratic function f ( x ) = x 2 − 6 x + 13 , we have a = 1 and b = − 6 . So, x = − 2 ( 1 ) − 6 = 2 6 = 3 . Now, we plug this value into the function to find the y-coordinate: f ( 3 ) = ( 3 ) 2 − 6 ( 3 ) + 13 = 9 − 18 + 13 = 4 . Thus, the vertex is ( 3 , 4 ) .
Final Answer The vertex of the parabola f ( x ) = x 2 − 6 x + 13 is ( 3 , 4 ) . Therefore, the correct answer is C.
Examples
Understanding the vertex of a parabola is very useful in real-world applications. For example, if you're launching a projectile, like a ball, the path it follows can be modeled by a parabola. The vertex of this parabola represents the highest point the ball will reach. Knowing this, you can calculate the maximum height and the distance at which it occurs, which is crucial in sports like basketball or baseball. Similarly, in engineering, when designing arched structures like bridges, understanding the vertex helps in determining the maximum load the structure can bear and ensuring its stability.
