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 (1)
To rewrite the quadratic equation y = 9 x 2 + 9 x − 1 in vertex form:
Factor out 9 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 as a squared term: y = 9 (( x + 2 1 ) 2 − 4 1 ) − 1 .
Simplify to obtain the vertex form: y = 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 asked 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 ) is the vertex of the parabola. Our goal is to complete the square to transform the given equation into vertex form.
Factoring First, factor out the coefficient of the x 2 term (which is 9) from the first two terms of the equation: y = 9 ( x 2 + x ) − 1
Completing the Square Next, we complete the square for the expression inside the parenthesis, x 2 + x . To do this, we take half of the coefficient of the x term, which is 2 1 , and square it, which gives us ( 2 1 ) 2 = 4 1 . We add and subtract this value inside the parenthesis to complete the square: y = 9 ( x 2 + x + 4 1 − 4 1 ) − 1
Rewriting as a Square Now, we rewrite the expression inside the parenthesis as a squared term: y = 9 ( ( x + 2 1 ) 2 − 4 1 ) − 1
Distributing Distribute the 9 to both terms inside the parenthesis: y = 9 ( x + 2 1 ) 2 − 9 ( 4 1 ) − 1
Simplifying Simplify the expression by multiplying: y = 9 ( x + 2 1 ) 2 − 4 9 − 1
Combining Constants Finally, combine the constant terms: y = 9 ( x + 2 1 ) 2 − 4 9 − 4 4 = 9 ( x + 2 1 ) 2 − 4 13 So, the vertex form of the given quadratic equation is y = 9 ( x + 2 1 ) 2 − 4 13 .
Final Answer Comparing our result with the given options, we find that the correct vertex form 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 can be modeled by a quadratic equation. By converting this equation to vertex form, you can easily find the maximum height the ball reaches and the time at which it reaches that height. This helps in predicting the trajectory and optimizing performance in sports.
