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=9\left(x+\frac{1}{2}\right)^2+\frac{5}{4}
y=9\left(x+\frac{1}{2}\right)^2-\frac{5}{4}
Answer (2)
The equation y = 9 x 2 + 9 x − 1 rewritten in vertex form is y = 9 ( x + 2 1 ) 2 − 4 13 . This transformation involves factoring out the leading coefficient, completing the square, and combining constant terms. The correct answer is the option: y = 9 ( x + 2 1 ) 2 − 4 13 .
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Factor out the coefficient of x 2 from the first two terms: y = 9 ( x 2 + x ) − 1 .
Complete the square inside the parenthesis: y = 9 ( x 2 + x + 4 1 − 4 1 ) − 1 .
Rewrite the expression as a squared term and distribute: y = 9 (( x + 2 1 ) 2 − 4 1 ) − 1 = 9 ( x + 2 1 ) 2 − 4 9 − 1 .
Combine the constant terms: y = 9 ( x + 2 1 ) 2 − 4 13 . The equation in vertex form is \boxed{y=9\tleft(x+\frac{1}{2}\right)^2-\frac{13}{4}} .
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, which is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.
Factoring To rewrite the equation in vertex form, we need to complete the square. First, factor out the coefficient of the x 2 term (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. Take half of the coefficient of the x term (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 Squared Term Rewrite the expression inside the parenthesis as a squared term:
y = 9 (( x + 2 1 ) 2 − 4 1 ) − 1
Distributing Distribute the 9:
y = 9 ( x + 2 1 ) 2 − 9 ( 4 1 ) − 1
Simplifying Simplify:
y = 9 ( x + 2 1 ) 2 − 4 9 − 1
Combining Constants Combine the constant terms. We have − 4 9 − 1 = − 4 9 − 4 4 = − 4 13 . So the equation becomes:
y = 9 ( x + 2 1 ) 2 − 4 13
Final Answer Therefore, the equation in vertex form is y = 9 ( x + 2 1 ) 2 − 4 13 .
Examples
Vertex form is useful in physics to describe the trajectory of a projectile, such as a ball thrown in the air. The vertex form of a quadratic equation helps determine the maximum height the ball reaches and the time at which it reaches that height. For example, if the height of a ball is given by h ( t ) = − 16 t 2 + 80 t + 5 , converting this to vertex form h ( t ) = − 16 ( t − 2.5 ) 2 + 105 tells us that the maximum height is 105 feet, reached at 2.5 seconds.
