The standard form of the equation of a parabola is $y=7 x^2+14 x+4$. What is the vertex form of the equation?
A. $y=7(x+2)^2+3$
B. $y=7(x+2)^2-3$
C. $y=7(x+1)^2-3$
D. $y=7(x+1)^2+3$
Answer (1)
Factor out the coefficient of x 2 : y = 7 ( x 2 + 2 x ) + 4 .
Complete the square: y = 7 ( x 2 + 2 x + 1 − 1 ) + 4 = 7 (( x + 1 ) 2 − 1 ) + 4 .
Distribute and simplify: y = 7 ( x + 1 ) 2 − 7 + 4 = 7 ( x + 1 ) 2 − 3 .
The vertex form is y = 7 ( x + 1 ) 2 − 3 .
Explanation
Problem Analysis We are given the equation of a parabola in standard form: y = 7 x 2 + 14 x + 4 . Our goal is to convert this equation into vertex form, which is given by y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola.
Factoring To convert the given equation to vertex form, we will use the method of completing the square. First, factor out the coefficient of the x 2 term (which is 7) from the first two terms of the equation: y = 7 ( x 2 + 2 x ) + 4
Completing the Square Now, we need to complete the square for the expression inside the parentheses, x 2 + 2 x . To do this, we take half of the coefficient of the x term (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and 1 2 = 1 . So, we add and subtract 1 inside the parentheses: y = 7 ( x 2 + 2 x + 1 − 1 ) + 4
Rewriting the Equation Rewrite the expression inside the parentheses as a squared term: y = 7 (( x + 1 ) 2 − 1 ) + 4
Distributing Now, distribute the 7 to both terms inside the parentheses: y = 7 ( x + 1 ) 2 − 7 + 4
Simplifying Finally, simplify the equation by combining the constant terms: y = 7 ( x + 1 ) 2 − 3
Final Answer The equation is now in vertex form: y = 7 ( x + 1 ) 2 − 3 . Comparing this to the given options, we see that it matches option C. Therefore, the vertex form of the given equation is y = 7 ( x + 1 ) 2 − 3 .
Examples
Understanding the vertex form of a parabola can help us analyze projectile motion. For example, if we launch a ball, its height over time can be modeled by a parabolic equation. The vertex form allows us to easily find the maximum height the ball reaches and the time at which it reaches that height. Suppose the height of the ball is given by y = − 4.9 t 2 + 9.8 t + 1 , where y is the height in meters and t is the time in seconds. Converting this to vertex form, we get y = − 4.9 ( t − 1 ) 2 + 5.9 . This tells us that the maximum height of the ball is 5.9 meters, and it reaches this height at t = 1 second.
