Which equation is $y=-6 x^2+3 x+2$ rewritten in vertex form?
A. $y=-6(x-1)^2+8$
B. $y=-6\left(x+\frac{1}{4}\right)^2+\frac{13}{8}$
C. $y=-6\left(x-\frac{1}{4}\right)^2+\frac{19}{8}$
D. $y=-6\left(x-\frac{1}{2}\right)^2+\frac{7}{2}$
Answer (1)
Factor out -6 from the x 2 and x terms: y = − 6 ( x 2 − 2 1 x ) + 2 .
Complete the square inside the parenthesis: y = − 6 ( x 2 − 2 1 x + 16 1 − 16 1 ) + 2 = − 6 (( x − 4 1 ) 2 − 16 1 ) + 2 .
Distribute the -6: y = − 6 ( x − 4 1 ) 2 + 16 6 + 2 = − 6 ( x − 4 1 ) 2 + 8 3 + 2 .
Combine the constants: y = − 6 ( x − 4 1 ) 2 + 8 19 . The equation in vertex form is y = − 6 ( x − 4 1 ) 2 + 8 19 .
Explanation
Understanding the Problem We are given the quadratic equation y = − 6 x 2 + 3 x + 2 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.
Factoring To convert the given equation to vertex form, we need to complete the square. First, factor out the coefficient of the x 2 term (which is -6) from the first two terms: y = -6\[x^2 - \frac{1}{2}x\] + 2
Completing the Square Now, we complete the square inside the parentheses. Take half of the coefficient of the x term, which is − 2 1 , so half of it is − 4 1 . Square this value to get ( − 4 1 ) 2 = 16 1 . Add and subtract this value inside the parentheses: y = − 6 ( x 2 − 2 1 x + 16 1 − 16 1 ) + 2
Rewriting as a Squared Term Rewrite the expression inside the parentheses as a squared term: y = -6\left(\[x - \frac{1}{4}\]^2 - \frac{1}{16}\right) + 2
Distributing Distribute the -6: y = − 6 ( x − 4 1 ) 2 + 16 6 + 2
Simplifying Simplify the constant term: y = − 6 ( x − 4 1 ) 2 + 8 3 + 2 To add the fractions, we need a common denominator, which is 8: y = − 6 ( x − 4 1 ) 2 + 8 3 + 8 16
Combining Constants Combine the constant terms: y = − 6 ( x − 4 1 ) 2 + 8 19
Final Answer Comparing our result with the given options, we find that the equation in vertex form is: y = − 6 ( x − 4 1 ) 2 + 8 19
Examples
Vertex form is useful in physics to describe the trajectory of a projectile, such as a ball thrown in the air. The equation y = − 6 ( x − 4 1 ) 2 + 8 19 models the height ( y ) of the ball at a horizontal distance ( x ). The vertex ( 4 1 , 8 19 ) represents the highest point the ball reaches, making it easy to find the maximum height and where it occurs.
