Write $y=-3 x^2-24 x-46$ in vertex form.
A. $y=-3(x-24)^2-46$
B. $y=8(x+16)^2+2$
C. $y=-3(x+4)^2+2$
D. $y=(x+4)^2-46$
Answer (1)
Factor out the coefficient of the x 2 term: y = − 3 ( x 2 + 8 x ) − 46 .
Complete the square: y = − 3 ( x 2 + 8 x + 16 − 16 ) − 46 = − 3 (( x + 4 ) 2 − 16 ) − 46 .
Distribute and simplify: y = − 3 ( x + 4 ) 2 + 48 − 46 = − 3 ( x + 4 ) 2 + 2 .
The vertex form is y = − 3 ( x + 4 ) 2 + 2 .
Explanation
Understanding the Problem We are given the quadratic equation y = − 3 x 2 − 24 x − 46 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 First, factor out the coefficient of the x 2 term (which is -3) from the first two terms: y = − 3 ( x 2 + 8 x ) − 46
Completing the Square Next, complete the square inside the parentheses. To do this, take half of the coefficient of the x term (which is 8), square it (which gives 16), and add and subtract it inside the parentheses: y = − 3 ( x 2 + 8 x + 16 − 16 ) − 46
Rewriting as a Squared Term Now, rewrite the expression inside the parentheses as a squared term: y = − 3 (( x + 4 ) 2 − 16 ) − 46
Distributing Distribute the -3: y = − 3 ( x + 4 ) 2 + 48 − 46
Simplifying Finally, simplify: y = − 3 ( x + 4 ) 2 + 2 The vertex form of the equation is y = − 3 ( x + 4 ) 2 + 2 .
Examples
Vertex form is useful in physics to determine, for example, the maximum height of a projectile. If the height of a ball thrown into the air is given by h ( t ) = − 3 t 2 + 6 t + 1 , we can rewrite this in vertex form to easily find the maximum height. Completing the square gives h ( t ) = − 3 ( t − 1 ) 2 + 4 , so the maximum height is 4 units, which occurs at time t=1.
