The vertex form of the equation of a parabola is $y=2(x+4)^2-21$. What is the standard form of the equation?
A. $y=2 x^2+16 x+11$
B. $y=2 x^2+12 x+17$
C. $y=4 x^2+4 x+9$
D. $x=2 y^2+5 y+17$
Answer (1)
Expand the squared term: ( x + 4 ) 2 = x 2 + 8 x + 16 .
Substitute back into the equation: y = 2 ( x 2 + 8 x + 16 ) − 21 .
Distribute the constant: y = 2 x 2 + 16 x + 32 − 21 .
Simplify to standard form: y = 2 x 2 + 16 x + 11 . The answer is y = 2 x 2 + 16 x + 11 .
Explanation
Understanding the Problem We are given the vertex form of a parabola's equation: y = 2 ( x + 4 ) 2 − 21 . Our goal is to convert this to standard form, which looks like y = a x 2 + b x + c .
Expanding the Squared Term First, we need to expand the squared term ( x + 4 ) 2 . Recall that ( x + a ) 2 = x 2 + 2 a x + a 2 . Therefore, ( x + 4 ) 2 = x 2 + 2 ( 4 ) x + 4 2 = x 2 + 8 x + 16 .
Substituting Back into the Equation Now, substitute this back into the original equation: y = 2 ( x 2 + 8 x + 16 ) − 21 .
Distributing the Constant Next, distribute the 2: y = 2 x 2 + 16 x + 32 − 21 .
Simplifying to Standard Form Finally, combine the constant terms: y = 2 x 2 + 16 x + 11 . This is the standard form of the equation.
Identifying the Correct Option Comparing our result y = 2 x 2 + 16 x + 11 with the given options, we see that it matches option A.
Examples
Understanding quadratic equations is crucial in various real-world applications. For instance, designing the trajectory of a projectile, such as a ball thrown in the air, involves quadratic functions. The height of the ball at any given time can be modeled by a quadratic equation, allowing us to predict its path and landing point. Similarly, engineers use quadratic equations to design parabolic mirrors and antennas, ensuring that they focus light or radio waves efficiently. These applications highlight the practical significance of converting between different forms of quadratic equations to solve real-world problems.
