Select the correct answer.
Consider this system of equations, where function [tex]$f$[/tex] is quadratic and function [tex]$g$[/tex] is linear:
[tex]
\begin{array}{l}
y=f(x) \\
y=g(x)
\end{array}
[/tex]
Which statement describes the number of possible solutions to the system?
A. The system may have no, 1, or 2 solutions.
B. The system may have 1 or 2 solutions.
C. The system may have no, 1, or infinite solutions.
D. The system may have no, 1, 2, or infinite solutions.
Answer (1)
The system of equations, consisting of a quadratic and a linear function, can have zero, one, or two solutions, corresponding to the number of intersection points between a parabola and a straight line. The correct answer is A. The system may have no, 1 , or 2 solutions. $\boxed{A}
Explanation
Understanding the Problem The problem asks us to determine the number of possible solutions for a system of equations where one equation is a quadratic function and the other is a linear function. Graphically, the solutions to this system are the points where the graphs of the two functions intersect.
Visualizing the Intersection A quadratic function forms a parabola, and a linear function forms a straight line. A straight line can intersect a parabola at zero, one, or two points.
Considering Possible Intersections If the line does not intersect the parabola, there are no solutions. If the line is tangent to the parabola, there is one solution. If the line intersects the parabola at two distinct points, there are two solutions.
Concluding the Number of Solutions Therefore, the system may have no, 1, or 2 solutions.
Examples
Imagine you're designing a roller coaster. The track's height can be modeled by a quadratic function, and a safety bar's height is represented by a linear function. The points where these functions intersect determine if the safety bar will interfere with the track. Understanding the possible number of solutions (0, 1, or 2) helps engineers ensure the roller coaster is both thrilling and safe.
