Write $y=-4 x^2-16 x-14$ in vertex form.
Answer (1)
Factor out the coefficient of the x 2 term: y = − 4 ( x 2 + 4 x ) − 14 .
Complete the square inside the parentheses: y = − 4 ( x 2 + 4 x + 4 − 4 ) − 14 .
Rewrite as a squared term and distribute: y = − 4 (( x + 2 ) 2 − 4 ) − 14 = − 4 ( x + 2 ) 2 + 16 − 14 .
Simplify to obtain the vertex form: y = − 4 ( x + 2 ) 2 + 2 , so the answer is y = − 4 ( x + 2 ) 2 + 2 .
Explanation
Understanding the Problem We are given the quadratic equation y = − 4 x 2 − 16 x − 14 and asked to write 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 -4) from the first two terms: y = − 4 ( x 2 + 4 x ) − 14
Completing the Square Now, complete the square inside the parentheses. Take half of the coefficient of the x term (which is 4), square it (which is ( 4/2 ) 2 = 2 2 = 4 ), and add and subtract it inside the parentheses: y = − 4 ( x 2 + 4 x + 4 − 4 ) − 14
Rewriting as a Squared Term Rewrite the expression inside the parentheses as a squared term: y = − 4 (( x + 2 ) 2 − 4 ) − 14
Distributing Distribute the -4: y = − 4 ( x + 2 ) 2 + 16 − 14
Simplifying Simplify: y = − 4 ( x + 2 ) 2 + 2
Final Answer The vertex form of the equation is y = − 4 ( x + 2 ) 2 + 2 .
Examples
Vertex form is useful in physics to describe projectile motion. For example, if the equation represents the height of a ball thrown in the air, the vertex form directly tells us the maximum height the ball reaches and the time at which it reaches that height. Understanding vertex form helps in optimizing trajectories and understanding the behavior of objects under gravity.
