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. $y=-6\left(x-\frac{1}{4}\right)^2+\frac{18}{8}$
D. $y=-6\left(x-\frac{1}{2}\right)^2+\frac{7}{2}$
Answer (2)
To rewrite the quadratic equation y = − 6 x 2 + 3 x + 2 in vertex form:
Factor out -6 from the first two terms: y = − 6 ( x 2 − f r a c 1 2 x ) + 2 .
Complete the square: y = − 6 ( x 2 − f r a c 1 2 x + f r a c 1 16 − f r a c 1 16 ) + 2 .
Rewrite as a perfect square: y = − 6 (( x − f r a c 1 4 ) 2 − f r a c 1 16 ) + 2 .
Simplify to get the vertex form: y = − 6 ( x − f r a c 1 4 ) 2 + f r a c 19 8 .
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. 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 -6) from the first two terms of the equation: y = − 6 ( x 2 − 2 1 x ) + 2
Completing the Square Next, we complete the square inside the parentheses. To do this, we take half of the coefficient of the x term, which is − 2 1 , and square it. Half of − 2 1 is − 4 1 , and squaring it gives ( − 4 1 ) 2 = 16 1 . We add and subtract this value inside the parentheses: y = − 6 ( x 2 − 2 1 x + 16 1 − 16 1 ) + 2
Rewriting as a Perfect Square Now, we rewrite the expression inside the parentheses as a perfect square: y = − 6 ( ( x − 4 1 ) 2 − 16 1 ) + 2
Distributing Distribute the -6 to both terms inside the parentheses: y = − 6 ( x − 4 1 ) 2 + 16 6 + 2
Simplifying Simplify the expression by finding a common denominator for the constants: y = − 6 ( x − 4 1 ) 2 + 8 3 + 8 16 y = − 6 ( x − 4 1 ) 2 + 8 19
Final Answer The equation in vertex form is y = − 6 ( x − 4 1 ) 2 + 8 19 . Comparing this to the given options, we see that the correct answer is y = − 6 ( x − 4 1 ) 2 + 8 19 . Note that 8 19 is not equal to 8 18 , so the third option is incorrect.
Examples
Vertex form is useful in physics to determine the maximum height of a projectile. For example, if the equation represents the height of a ball thrown in the air, the vertex form immediately tells us the maximum height the ball reaches and the time at which it reaches that height. This is because the vertex (h, k) represents the highest or lowest point of the parabola, depending on whether the parabola opens upwards or downwards. In engineering, vertex form can help optimize designs, such as finding the optimal curvature for a bridge to maximize strength and minimize material usage. It also helps in economics to find maximum profit or minimum cost scenarios.
The equation y = − 6 x 2 + 3 x + 2 rewritten in vertex form is y = − 6 ( x − 4 1 ) 2 + 8 19 . Upon reviewing the options provided, none correspond directly to the derived vertex form. This suggests the original options may not fully align with the correct transformation.
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