The vertex form of the equation of a parabola is $y=(x-3)^2+36$. What is the standard form of the equation?
A. $y=x^2-6 x+45$
B. $y=x^2+x+18$
C. $y=3 x^2-6 x+45$
D. $y=x^2+6 x+36$
Answer (1)
Expand the vertex form: y = ( x − 3 ) 2 + 36 .
Expand the squared term: ( x − 3 ) 2 = x 2 − 6 x + 9 .
Substitute and simplify: y = x 2 − 6 x + 9 + 36 .
Combine constants to get the standard form: y = x 2 − 6 x + 45 .
Explanation
Understanding the Problem We are given the vertex form of a parabola equation: y = ( x − 3 ) 2 + 36 . Our goal is to convert this to standard form, which is y = a x 2 + b x + c .
Expanding the Squared Term To convert from vertex form to standard form, we need to expand and simplify the given equation. First, let's expand the squared term: ( x − 3 ) 2 = ( x − 3 ) ( x − 3 ) = x 2 − 3 x − 3 x + 9 = x 2 − 6 x + 9
Substituting Back into the Equation Now, substitute this back into the original equation: y = ( x 2 − 6 x + 9 ) + 36
Combining Constant Terms Next, combine the constant terms: y = x 2 − 6 x + 9 + 36 = x 2 − 6 x + 45
Final Answer So, the standard form of the equation is y = x 2 − 6 x + 45 . Comparing this 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 standard form helps in easily identifying the coefficients that determine the shape and position of the parabola, allowing engineers and physicists to predict the projectile's path accurately. Similarly, quadratic equations are used in optimizing areas and volumes in construction and manufacturing.
