What is the vertex form of the equation of a parabola, given that the standard form is [tex]$y=x^2-8 x+29$[/tex]?
A. [tex]$y=(x-4)^2+18$[/tex]
B. [tex]$y=(x-4)^2+13$[/tex]
C. [tex]$y=(x-4)^2+28$[/tex]
D. [tex]$y=(x+4)(x-4)+13$[/tex]
Answer (1)
Identify the coefficient of the x term in the standard form equation: y = x 2 − 8 x + 29 .
Calculate the value to complete the square: ( 2 − 8 ) 2 = 16 .
Add and subtract this value within the equation and rewrite as a squared term: y = ( x − 4 ) 2 − 16 + 29 .
Simplify to obtain the vertex form: y = ( x − 4 ) 2 + 13 , so the final answer is y = ( x − 4 ) 2 + 13 .
Explanation
Understanding the Problem We are given the standard form of a parabola's equation: y = x 2 − 8 x + 29 . Our goal is to convert this into vertex form, which is y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola.
Completing the Square To convert the given equation to vertex form, we'll use the method of completing the square. This involves manipulating the equation to create a perfect square trinomial.
Calculating the Value to Complete the Square First, focus on the x 2 − 8 x terms. We need to find a value to add and subtract to complete the square. Take half of the coefficient of the x term (which is -8) and square it: ( 2 − 8 ) 2 = ( − 4 ) 2 = 16 .
Adding and Subtracting the Value Now, add and subtract 16 within the equation: y = x 2 − 8 x + 16 − 16 + 29 .
Rewriting as a Squared Term Rewrite the first three terms as a squared term: y = ( x − 4 ) 2 − 16 + 29 .
Simplifying the Equation Simplify the constant terms: y = ( x − 4 ) 2 + 13 .
Final Answer The vertex form of the equation is y = ( x − 4 ) 2 + 13 . Therefore, the vertex of the parabola is ( 4 , 13 ) .
Examples
Vertex form is incredibly useful in physics, especially when analyzing projectile motion. For example, if you kick a ball, the height of the ball over time follows a parabolic path. The vertex form of the parabola tells you the maximum height the ball reaches and when it reaches that height. This is also applicable in engineering when designing arches or suspension bridges, where understanding the vertex helps ensure structural stability.
