Where a Line Meets a Parabola

Two graphs on the same axes, and the question is where they cross. It comes up whenever two situations are being compared — a cost that grows steadily against one that accelerates, a projectile against a slope, a revenue line against a cost curve.

The crossing points are the values of $x$ where the two functions agree, so setting them equal is not a trick. It is the definition.

$\text{Solve }f(x)=g(x)\text{where}f(x)=x^2-2x-3,g(x)=x+1$

Bring everything to one side, and a familiar object appears:

$x^2-2x-3=x+1⟹x^2-3x-4=0⟹(x-4)(x+1)=0$

So $x=4$ or $x=-1$. Substitute back into the simpler function — the line, always — to get the points: $(4,5)$ and $(-1,0)$.

The three cases, and what they mean

Discriminant of the combined equation	Graph	The situation
$b^2-4ac>0$	Two intersection points	The line cuts the parabola
$b^2-4ac=0$	One point	The line is tangent — it touches and leaves
$b^2-4ac<0$	No points	They never meet; no real solution

This is the same discriminant from Quadratic Functions Revisited, doing a new job. When a question asks for the value of $k$ that makes a line tangent to a curve, it is asking you to set the discriminant to zero and solve — a question that looks impossible until you notice that.

Graphically, and why you should do both

Plot both in Using Desmos before solving. The graph tells you how many solutions to expect and roughly where, which catches the two most common errors instantly: a sign slip that produces two answers where the picture shows none, and an arithmetic slip that puts a crossing point in the wrong quadrant.

The graph is not the answer, though. Reading “about 3.9” off a screen is not solving; the algebra gives $x=4$ exactly, and exact is what a later calculation needs.

Substitute into the easier one

Once you have the $x$ values, put them into whichever function is simpler — usually the line. Students routinely substitute into the quadratic, do more arithmetic than necessary, and make a mistake in it. Both functions must give the same $y$; that is what “intersection” means, and checking one against the other is a free verification.

Curriculum connection

A2.5

solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determine the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being filled at a constant rate and the other is being emptied through a hole in the bottom) Sample problem: Determine, through investigation, the equations of the lines that have a slope of 2 and that intersect the quadratic function $f(x)=x(6-x)$ once; twice; never.

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A2.4

determine, through investigation, the transformational relationship among the family of quadratic functions that have the same zeros, and determine the algebraic representation of a quadratic function, given the real roots of the corresponding quadratic equation and a point on the function Sample problem: Determine the equation of the quadratic function that passes through $(2,5)$ if the roots of the corresponding quadratic equation are $1+\sqrt{5}$ and $1-\sqrt{5}$.

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