For a coordinate distance question, the requested length is the straight segment between two points. A line and a parabola can meet twice, so read both intersections from the graph. Their horizontal and vertical changes form a right triangle. If those changes are equal, use the -- triangle shortcut. One change alone is a side of the triangle, not the requested distance.
Hints
- Hint 1
An intersection lies on both graphs, so it solves the system. Read the coordinates at each place the line crosses the parabola. What two points do you see?
- Hint 2
Measure the horizontal change by subtracting the -coordinates, then measure the vertical change by subtracting the -coordinates. These changes form two sides that meet at a right angle.
- Hint 3
If the two perpendicular sides have the same length, they form a -- triangle. Its longest side, called the hypotenuse, is times either shorter side.
Step-by-step
Read the intersections and form a right triangle
Step 1Find the two solutions on the graph
No Desmos needed. Read the intersection points directly from the graph. The line and parabola share the grid points and . Each shared point is a solution because it lies on both graphs.
- Step 2
Find the horizontal change
From to , subtract the -coordinates to find the horizontal change:
- Step 3
Find the vertical change
From to , subtract the -coordinates to find the vertical change:
- Step 4
Find the distance between the points
The horizontal and vertical changes meet at a right angle, so they are equal legs of a -- triangle. The distance between the intersections is the hypotenuse, not a leg. A triangle of this kind has a hypotenuse times either leg. Multiply the leg length by :
So the distance between the two solutions is . Choice C.