Write $y=2 x^2-12 x+16$ in vertex form.
A. $y=2(x-3)^2-2$
B. $y=-6(x+9)^2-2$
C. $y=(x-3)^2+16$
D. $y=2(x-12)^2+16$
Answer (1)
To convert the quadratic equation to vertex form:
Factor out the leading coefficient from the quadratic and linear terms: y = 2 ( x 2 − 6 x ) + 16 .
Complete the square inside the parenthesis by adding and subtracting ( 2 − 6 ) 2 = 9 : y = 2 ( x 2 − 6 x + 9 − 9 ) + 16 .
Rewrite the quadratic as a squared term: y = 2 (( x − 3 ) 2 − 9 ) + 16 .
Simplify to obtain the vertex form: y = 2 ( x − 3 ) 2 − 2 .
Explanation
Understanding the Problem We are given the quadratic equation y = 2 x 2 − 12 x + 16 and we want to write it in vertex form, which is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.
Factoring First, factor out the coefficient of the x 2 term (which is 2) from the first two terms of the equation: y = 2 ( x 2 − 6 x ) + 16
Completing the Square To complete the square for the expression inside the parentheses, we need to add and subtract the square of half of the coefficient of the x term. The coefficient of the x term is -6, so half of it is -3, and the square of -3 is ( − 3 ) 2 = 9 . Thus, we add and subtract 9 inside the parentheses: y = 2 ( x 2 − 6 x + 9 − 9 ) + 16
Rewriting as a Squared Term Now, rewrite the expression inside the parentheses as a squared term: y = 2 (( x − 3 ) 2 − 9 ) + 16
Distributing Distribute the 2: y = 2 ( x − 3 ) 2 − 18 + 16
Simplifying Finally, simplify the equation: y = 2 ( x − 3 ) 2 − 2 So, the vertex form of the given quadratic equation is y = 2 ( x − 3 ) 2 − 2 .
Examples
Vertex form is incredibly useful in physics, especially when analyzing projectile motion. For example, if you're analyzing the height of a ball thrown into the air, the vertex form of the height equation will immediately tell you the maximum height the ball reaches and the time at which it reaches that height. This makes understanding and predicting the behavior of objects in motion much easier.
