The standard form of the equation of a parabola is $x=y^2+6 y+1$. What is the vertex form of the equation?
A. $x=(y+6)^2-11$
B. $x=(y+3)^2-5$
C. $x=(y+3)^2-8$
D. $x=(y+6)^2-35$
Answer (2)
Complete the square for the y terms: x = y 2 + 6 y + 1 = ( y 2 + 6 y + 9 ) − 9 + 1 .
Rewrite the squared term: x = ( y + 3 ) 2 − 9 + 1 .
Simplify the constants: x = ( y + 3 ) 2 − 8 .
The vertex form of the equation is x = ( y + 3 ) 2 − 8 .
Explanation
Understanding the Problem We are given the equation of a parabola in standard form: x = y 2 + 6 y + 1 . Our goal is to convert this equation to vertex form, which looks like x = ( y − k ) 2 + h , where ( h , k ) represents the vertex of the parabola. To do this, we'll complete the square for the expression involving y .
Completing the Square To complete the square for y 2 + 6 y + 1 , we need to focus on the y 2 + 6 y part. We want to find a constant term to add and subtract so that we can rewrite this as a squared term. Take half of the coefficient of the y term (which is 6), square it, and add and subtract it within the equation. Half of 6 is 3, and 3 2 = 9 . So, we add and subtract 9:
x = y 2 + 6 y + 9 − 9 + 1
Rewriting in Vertex Form Now, we can rewrite the first three terms as a squared term:
x = ( y + 3 ) 2 − 9 + 1
Combine the constants:
x = ( y + 3 ) 2 − 8
Final Answer The equation is now in vertex form: x = ( y + 3 ) 2 − 8 . Comparing this to the options given, we see that it matches option C.
Therefore, the vertex form of the equation is x = ( y + 3 ) 2 − 8 .
Examples
Understanding parabolas is crucial in various fields. For example, engineers use parabolic shapes in designing satellite dishes and radio telescopes to focus signals at a single point. Architects use parabolic arches for structural support in bridges and buildings. In sports, the trajectory of a ball thrown or kicked often approximates a parabola, which athletes and coaches consider to optimize performance.
The vertex form of the given equation x = y 2 + 6 y + 1 is x = ( y + 3 ) 2 − 8 , matching option C. This was determined by completing the square for the y terms. The resulting vertex of the parabola is at the point ( − 8 , − 3 ) .
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