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Math foundations
SAT nonlinear systems: where a line meets a parabola
A line can cross a parabola twice, touch it once, or miss it completely. See all three cases on one graph, then solve by substitution and count solutions with the discriminant.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- Solve a line-and-parabola system by setting the expressions for y equal and solving the quadratic.
- A system of a line and a parabola can have 0, 1, or 2 solutions.
- The discriminant of the combined quadratic counts the solutions without solving.
In this article
Set the two expressions for y equal
Consider \(\displaystyle y = x^2 - 4x + 3\) and \(\displaystyle y = x - 1\). At an intersection, both equations give the same y, so set them equal: \(\displaystyle x^2 - 4x + 3 = x - 1\). Move everything to one side: \(\displaystyle x^2 - 5x + 4 = 0\), which factors as \(\displaystyle (x - 1)(x - 4) = 0\).
So \(\displaystyle x = 1\) or \(\displaystyle x = 4\). Substitute into the simpler equation, \(\displaystyle y = x - 1\), to get the points (1, 0) and (4, 3).
The study lab
A line crossing a parabola twice
y = x² − 4x + 3 and y = x − 1 intersect at (1, 0) and (4, 3).
- y = x² − 4x + 3
- y = x − 1
Two, one, or zero solutions
The graph below shows \(\displaystyle y = x^2\) with three parallel lines. The line \(\displaystyle y = 2x\) crosses the parabola twice. The line \(\displaystyle y = 2x - 1\) just touches it at (1, 1). The line \(\displaystyle y = 2x - 3\) misses it completely.
The study lab
One parabola, three lines
y = x² with y = 2x (two solutions), y = 2x − 1 (one), and y = 2x − 3 (none).
- y = x²
- y = 2x (2 solutions)
- y = 2x − 1 (1 solution)
Horizontal lines and the vertex
When the line is horizontal, \(\displaystyle y = c\), the vertex tells you everything. A parabola that opens up with its vertex at height 4 touches \(\displaystyle y = 4\) once, crosses any higher horizontal line twice, and never reaches a lower one.
Your turn
Quick practice
For what value of k does the system y = x² + k and y = 4 have exactly one solution?
Hint: When does x² equal a number in exactly one way?
Practice: try these before checking the answers
Set the expressions equal, then solve. Graph in Desmos to check.
- 1. Solve the system \(\displaystyle y = x^2\) and \(\displaystyle y = x + 2\).
- 2. How many solutions do \(\displaystyle y = x^2 + 1\) and \(\displaystyle y = 0\) have?
- 3. Solve the system \(\displaystyle y = -x^2 + 6\) and \(\displaystyle y = 2\).
- 4. How many solutions do \(\displaystyle y = x^2 - 6x + 9\) and \(\displaystyle y = 0\) have?
Sources & further reading
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