A system containing a line and a curve can have zero, one, or multiple real solutions. Each solution is an ordered pair at which the graphs intersect and both equations are true.
Essential concepts
\[f(x)=g(x)\]
If y = f(x) and y = g(x), their intersections occur at inputs where the two outputs are equal.
SAT strategy and application
Find line-parabola intersections
Solve \(y=x+2\) and \(y=x^2-4\).
- Set the expressions equal
\(x+2=x^2-4\), so \(x^2-x-6=0\).
- Factor and evaluate
\((x-3)(x+2)=0\), giving \(x=3\) or \(x=-2\). Then \(y=x+2\).
Intersections: \((3,5)\) and \((-2,0)\).
Mistakes and traps
Check your understanding
If \(y=x^2\) and \(y=4\), what are the intersection x-coordinates?
- \(2\)
- \(-2\)
- \(pm2\)
- \(pm4\)
Show answer and explanation
Answer: \(pm2\)
\(x^2=4\) has solutions \(x=2\) and \(x=-2\).
Key takeaways
What to remember
- System solutions are intersections that satisfy every equation.
- Set function outputs equal to reduce the system to one variable.
- Recover and correctly pair y-values for every x-solution.
- Use graph behavior to check the likely number of solutions.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.