The center of an ice rink is located at $(0,0)$ on a coordinate system measured in feet. Sandi skates along a path that can be modeled by the equation $y=0.08 x^2-1.6 x+13$. David starts at $(20,13)$ and skates along a path that can be modeled by a quadratic function with a vertex at $(10,5)$.

Which system of equations models the paths of the two skaters?

A. $y=0.08 x^2-1.6 x+13$ and $y=0.08(x-10)^2+5$
B. $y=0.08 x^2-1.6 x+13$ and $y=1.6(x-10)^2+5$
C. $y=0.08 x^2-1.6 x+13$ and $y=0.8(x-10)^2+5$
D. $y=0.08 x^2-1.6 x+13$ and $y=5(x-10)^2+5$

The equation for Sandi's path is given as y = 0.08 x 2 − 1.6 x + 13 .
David's path is a quadratic function with vertex ( 10 , 5 ) , so its equation is of the form y = a ( x − 10 ) 2 + 5 .
Since David's path passes through ( 20 , 13 ) , substitute this point into the equation to find a : 13 = a ( 20 − 10 ) 2 + 5 , which simplifies to a = 0.08 .
Therefore, the system of equations is y = 0.08 x 2 − 1.6 x + 13 y = 0.08 ( x − 10 ) 2 + 5 ​ .

Explanation

Analyze the problem We are given the equation for Sandi's path as y = 0.08 x 2 − 1.6 x + 13 . We need to find the equation for David's path, which is a quadratic function with a vertex at ( 10 , 5 ) and passes through the point ( 20 , 13 ) .

Write the vertex form of the quadratic equation The general vertex form of a quadratic equation is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In this case, the vertex is ( 10 , 5 ) , so the equation for David's path is y = a ( x − 10 ) 2 + 5 .

Substitute the point (20,13) into the equation Since David's path passes through the point ( 20 , 13 ) , we can substitute these coordinates into the equation to solve for a :


13 = a ( 20 − 10 ) 2 + 5

Solve for a Now, we solve for a :

13 = a ( 10 ) 2 + 5
13 = 100 a + 5
8 = 100 a
a = 100 8 ​ = 0.08

Write the system of equations So, the equation for David's path is y = 0.08 ( x − 10 ) 2 + 5 . Therefore, the system of equations that models the paths of the two skaters is:

y = 0.08 x 2 − 1.6 x + 13 and y = 0.08 ( x − 10 ) 2 + 5

Final Answer The system of equations that models the paths of the two skaters is:

y = 0.08 x 2 − 1.6 x + 13 and y = 0.08 ( x − 10 ) 2 + 5
Examples
Understanding quadratic functions is crucial in various real-world scenarios, such as modeling the trajectory of a projectile, designing parabolic reflectors for satellite dishes, or optimizing the shape of an arch in architecture. In this problem, we use quadratic functions to model the paths of two skaters on an ice rink. By determining the equations that represent their paths, we can analyze their movements, predict their future positions, and even optimize their routes for specific performances. For example, if the skaters were to perform a routine where they meet at a certain point, we could use the system of equations to find the intersection point of their paths, ensuring a synchronized and visually appealing performance. The ability to model and analyze such scenarios using quadratic functions highlights the practical significance of this mathematical concept in various fields.

Answered by GinnyAnswer | 2025-07-05