Lesson 5 of 5
Find and check the shared points of a line and a parabola.
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Worked example
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Solve and . Check: and give the one ordered pair .
Big ideas
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A system solution is an ordered pair satisfying both equations at once. Its graph is an intersection of the line and parabola.
If and , set . Move all terms to one side, then solve the quadratic for x.
For each candidate x, calculate y from the line. Keep each matching pair together; giving only x-values does not completely solve a two-variable system.
A graph with a nonzero quadratic coefficient and a line can intersect at zero, one, or two points. A repeated root gives one tangent point. Check both originals.
Worked example
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Solve and . Equate and factor: . Find y from the line to get and . Check both points in the parabola.
Step-through
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Solve and .
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Worked example
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Solve and . Equate: . Thus and . There is one intersection , even though the root is repeated. Check and .
Worked example
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Solve and . Equating gives , impossible over the reals. There are no shared real ordered pairs: the parabola always has output at least 4.
Mix-up alert
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Wrong idea: and are a complete answer for a system in x and y. Repair: compute the corresponding y for each x, write both ordered pairs, and substitute each into both equations.
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Both equations must hold.
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What does a system solution represent on a graph?
A point shared by both graphs.
How do you combine and ?
Set .
What do you do after solving for x?
Find y for each x, keeping the coordinates paired.
How many intersections can and have when a is nonzero?
Zero, one, or two real intersections.
What does a repeated quadratic root mean here?
One intersection point, where the nonvertical line is tangent.
Why check both original equations?
A point must satisfy both to solve the system.
Solve and .
.
Can a vertical line meet twice?
No; its one shared point is .
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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