Question

Consider a system of equations formed by a line and a parabola. How many solutions are possible? Sketch a graph to show each possibility.

Answer 1

The answer of AL2006 in the comments section is correct. I would just show the sketch and elaborate the situations in which there are 0, 1 and 2 solutions.

A parabola is characterized by a function that looks like an arc when graphed. Thus, it always passes twice on either the x-axis or the y-axis. The parabola in the picture is the one in red. Now, when you draw a diagonal line like that shown in blue, you intersect the parabola twice. Thus, you have two solutions. If you draw a diagonal line that is tangent to the parabola, you have a one solution. A tangent line is a line that intersects with a parabola at one single point only, as that shown in purple. Lastly, if you draw a horizontal or vertical line that is not part of the domain or range of the parabola, you don't have any solution for the system because it will never intersect (shown in green).