Which equation is $y=9 x^2+9 x-1$ rewritten in vertex form?
y=9\left(x+\frac{1}{2}\right)^2-\frac{13}{4}
y=9\left(x+\frac{1}{2}\right)^2-1
y=8\left(x+\frac{1}{2}\right)^2+\frac{5}{4}
y=9\left(x+\frac{1}{2}\right)^2-\frac{5}{4}
Answer (2)
Factor out the coefficient of x 2 : y = 9 ( x 2 + x ) − 1 .
Complete the square: y = 9 ( x 2 + x + 4 1 − 4 1 ) − 1 = 9 (( x + 2 1 ) 2 − 4 1 ) − 1 .
Distribute and simplify: y = 9 ( x + 2 1 ) 2 − 4 9 − 1 = 9 ( x + 2 1 ) 2 − 4 13 .
The equation in vertex form is y = 9 ( x + 2 1 ) 2 − 4 13 .
Explanation
Understanding the Problem We are given the quadratic equation y = 9 x 2 + 9 x − 1 and we want to rewrite it in vertex form. The vertex form of a quadratic equation is given by y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola.
Factoring To convert the given equation to vertex form, we will complete the square. First, factor out the coefficient of x 2 (which is 9) from the first two terms:
y = 9 ( x 2 + x ) − 1
Completing the Square Now, we complete the square inside the parenthesis. To do this, we take half of the coefficient of x (which is 1), square it, and add and subtract it inside the parenthesis. Half of 1 is 2 1 , and ( 2 1 ) 2 = 4 1 . So we have:
y = 9 ( x 2 + x + 4 1 − 4 1 ) − 1
Rewriting as a Square Rewrite the expression inside the parenthesis as a square:
y = 9 (( x + 2 1 ) 2 − 4 1 ) − 1
Distributing Distribute the 9:
y = 9 ( x + 2 1 ) 2 − 9 ( 4 1 ) − 1
y = 9 ( x + 2 1 ) 2 − 4 9 − 1
Combining Constants Combine the constants:
y = 9 ( x + 2 1 ) 2 − 4 9 − 4 4
y = 9 ( x + 2 1 ) 2 − 4 13
Final Answer Therefore, the vertex form of the given quadratic equation is y = 9 ( x + 2 1 ) 2 − 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. Rewriting the equation of this path in vertex form allows you to easily find the maximum height the ball reaches and the time at which it reaches that height. This helps predict the trajectory and optimize the launch angle for maximum distance.
The vertex form of the equation y = 9 x 2 + 9 x − 1 is y = 9 ( x + 2 1 ) 2 − 4 13 . We achieved this by completing the square and simplifying. The chosen answer is the first option: y = 9 ( x + 2 1 ) 2 − 4 13 .
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