The vertex form of the equation of a parabola is $x=8(y-1)^2-15$. What is the standard form of the equation?
A. $x=8 y^2-6 y-22$
B. $x=8 y^2-16 y-7$
C. $x=16 y^2-4 y+32$
D. $x=16 y^2-2 y-16$
Answer (1)
Expand the vertex form: x = 8 ( y − 1 ) 2 − 15 .
Simplify the equation: x = 8 ( y 2 − 2 y + 1 ) − 15 = 8 y 2 − 16 y + 8 − 15 .
Combine constants: x = 8 y 2 − 16 y − 7 .
The standard form of the equation is x = 8 y 2 − 16 y − 7 .
Explanation
Understanding the Problem The given equation is in vertex form: x = 8 ( y − 1 ) 2 − 15 . We need to convert it to standard form, which is x = a y 2 + b y + c .
Expanding the Square First, expand the squared term: ( y − 1 ) 2 = y 2 − 2 y + 1 .
Substituting Back Now, substitute this back into the equation: x = 8 ( y 2 − 2 y + 1 ) − 15 .
Distributing Distribute the 8: x = 8 y 2 − 16 y + 8 − 15 .
Simplifying Finally, combine the constants: x = 8 y 2 − 16 y − 7 .
Finding the Answer Comparing this with the given options, we see that it matches option B.
Examples
Understanding parabolas is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The vertex form helps determine the maximum height and range of the projectile, while the standard form can be useful for analyzing the overall shape and position of the parabola in a coordinate system. This knowledge is essential for designing accurate targeting systems or understanding the behavior of objects under gravity.
