The graph: the parabola
Learning Objectives
- Plot a quadratic function point by point and name its graph: the parabola.
- Find the intersection of the parabola with \(Oy\) — always the single point \((0, c)\).
- Count and compute the intersections with \(Ox\) from the sign of \(\Delta\), using Unit 1.
- Read the concavity (opens upward or downward) from the sign of \(a\), and the width from \(|a|\).
- Sketch any quadratic function from vertex, concavity, axis points, and symmetry.
- Prove that two quadratic functions agreeing at three points are equal.
Time to draw
Lesson 1 of this unit told us a great deal about \(f(x) = ax^2 + bx + c\) without a single picture: a vertex, a lowest or highest value, a mirror symmetry. Now we assemble the picture itself. As always, the graph is the set of points
\[G_f = \{\,(x,\, f(x)) \mid x \in \mathbb{R}\,\}\]
in the Cartesian plane. Start with the simplest quadratic function, \(f(x) = x^2\), and plot values:
\[f(-3) = 9,\;\; f(-2) = 4,\;\; f(-1) = 1,\;\; f(0) = 0,\;\; f(1) = 1,\;\; f(2) = 4,\;\; f(3) = 9.\]
The points fall into a smooth, U-shaped curve — dipping to the vertex \(V(0, 0)\), perfectly mirror-symmetric about the \(y\)-axis, exactly as the theorems of Lesson 1 predicted. Unlike the line of the linear function, this curve bends everywhere: no part of it is straight.
The parabola
The graph of a quadratic function is called a parabola.
The vertex \(V(x_V, y_V)\) of the function is called the vertex of the parabola; the two halves of the curve on either side of the axis of symmetry \(x = x_V\) are its branches.
If \(a > 0\), the branches open upward — the vertex is the lowest point (the function's minimum, from Lesson 1).
If \(a < 0\), the branches open downward — the vertex is the highest point (the maximum).
This orientation of the branches is called the concavity of the parabola.
Intersections with the coordinate axes
Let \(f(x) = ax^2 + bx + c\) with graph \(G_f\).
\(G_f\) meets the \(Oy\) axis in exactly one point: \((0, c)\).
\(G_f\) meets the \(Ox\) axis according to the sign of \(\Delta = b^2 - 4ac\): if \(\Delta > 0\), in exactly two points \((x_1, 0)\) and \((x_2, 0)\), where \(x_{1,2}\) are the roots from Unit 1; if \(\Delta = 0\), in exactly one point — which is the vertex \(V(x_V, 0)\) itself; if \(\Delta < 0\), in no point at all.
A point lies on \(Oy\) exactly when its abscissa is \(0\); the only graph point with abscissa \(0\) is \((0, f(0))\), and \(f(0) = c\). (A function assigns one value to \(0\), never two — which is why every parabola crosses \(Oy\) exactly once.)
A point of \(G_f\) lies on \(Ox\) exactly when its ordinate vanishes: \(f(x) = 0\). This is Unit 1's equation, and its theorem does the counting: two solutions when \(\Delta > 0\), one when \(\Delta = 0\), none when \(\Delta < 0\).
Only the middle case needs one more look. When \(\Delta = 0\), the single solution is \(x = -\tfrac{b}{2a} = x_V\), and Lesson 1 of this unit showed \(y_V = -\tfrac{\Delta}{4a} = 0\): the unique intersection point is precisely the vertex. The parabola does not cross the axis there — it touches it at its lowest (or highest) point and turns back.
Sketching a parabola in four steps
To sketch \(f(x) = x^2 - 4x + 3\) — the function of Lesson 1's value table — collect what the theory hands us:
Concavity: \(a = 1 > 0\), so the branches open upward.
Vertex and axis: \(V(2, -1)\), axis of symmetry \(x = 2\) (computed in Lesson 1).
Axis intersections: \(f(0) = 3\) gives \((0, 3)\) on \(Oy\); and \(\Delta = 4 > 0\) with roots \(1\) and \(3\) gives \((1, 0)\) and \((3, 0)\) on \(Ox\).
Symmetry for free points: the mirror image of \((0, 3)\) across the axis \(x = 2\) is \((4, 3)\) — a fifth point at no computational cost.
Five points and the concavity determine the sketch completely; the smooth curve through them is the parabola.
The geometric aspect: what \(a\) does to the shape
All parabolas share the same U-form, but \(a\) tunes it. The sign of \(a\) picks the direction; the size \(|a|\) picks the width: a larger \(|a|\) magnifies the term \(a(x - x_V)^2\), so the branches climb (or fall) faster — a narrower, steeper parabola. A smaller \(|a|\) gives a wider, flatter one.
Note what \(b\) and \(c\) do not do: they slide the parabola around the plane (they move the vertex), but the shape itself — direction and width — is governed by \(a\) alone. That is visible in the canonical form \(f(x) = a(x - x_V)^2 + y_V\): the coefficients \(b\) and \(c\) hide inside \(x_V\) and \(y_V\), while \(a\) stands in front of the square.
Three points determine the parabola
If two quadratic functions \(f\) and \(g\) take equal values at three distinct inputs, then \(f = g\) — they are the same function, with the same graph.
In consequence: two distinct parabolas can share at most two points, and a parabola is completely pinned down by any three of its points. (Recall the line needed only two; the extra point pays for the extra coefficient.)
Let \(h(x) = f(x) - g(x)\), and let \(u_1, u_2, u_3\) be the three distinct inputs where \(f\) and \(g\) agree, so that \(h(u_1) = h(u_2) = h(u_3) = 0\). Expanding the two general forms, \(h\) is a polynomial expression of degree at most \(2\): \(h(x) = Ax^2 + Bx + C\) with \(A, B, C \in \mathbb{R}\). Go through the possibilities for \(h\).
If \(A \neq 0\): then \(h(x) = 0\) is a quadratic equation, and by Unit 1 it has at most two solutions — it cannot vanish at three distinct inputs.
If \(A = 0\) and \(B \neq 0\): then \(h(x) = Bx + C\) vanishes only at \(x = -\tfrac{C}{B}\), a single input — again too few.
If \(A = B = 0\) and \(C \neq 0\): then \(h\) never vanishes at all.
The only case left is \(A = B = C = 0\), that is, \(f(x) = g(x)\) for every real \(x\).
Connection to Computer Science
The letters you are reading right now are drawn from parabolas. Most font technology (TrueType, used on nearly every phone and computer) stores each character's outline as a chain of short parabolic arcs, each pinned down by three points — exactly the determination we just proved. When you change the text size, the computer re-evaluates those quadratic functions and redraws every letter perfectly smooth at the new scale; storing curves instead of pixels is what keeps fonts sharp at any size.
The eye is not a proof
Every parabola is a smooth U-shaped curve. Is every smooth U-shaped curve a parabola?
No — and the counterexamples are instructive. The graph of \(g(x) = x^4\) looks convincingly parabolic: symmetric, U-shaped, vertex at the origin. But no parabola passes through all its points. Our uniqueness theorem even supplies the test: the points \((0,0)\), \((1,1)\), \((-1,1)\) of this graph lie on exactly one parabola — namely \(y = x^2\) — and a fourth point of the graph, \((2, 16)\), misses it, since \(2^2 = 4 \neq 16\). Three points pin down the only candidate; a fourth point can convict it.
History offers a famous warning. A chain hanging freely between two posts traces a curve so parabola-like that Galileo himself asserted it was a parabola. Decades later the seventeen-year-old Christiaan Huygens proved it is not: the true curve of the hanging chain (now called a catenary) is described by functions beyond this course. The two curves agree closely near the bottom and part ways in the arms — invisibly to the eye, unmistakably to algebra.
The habit of rigor: a shape earns the name parabola only by exhibiting the quadratic function whose graph it is — the eye proposes, algebra disposes.
Exercises
Exercise 1
For \(f(x) = x^2 - 2x - 8\), find the intersection points of the parabola with the axes.
Exercise 2
Without sketching, determine how many points each parabola shares with the \(Ox\) axis.
Exercise 3
Decide whether each statement about parabolas is true or false.
Exercise 4
For \(f(x) = -x^2 + 2x + 3\) (branches open downward, since \(a = -1 < 0\)), collect the data for a sketch.
Exercise 5
Compare each parabola's width with that of \(y = x^2\), using \(|a|\).
Summary
The graph of a quadratic function is the parabola: vertex \(V(x_V, y_V)\), two branches, axis of symmetry \(x = x_V\); branches open upward for \(a > 0\), downward for \(a < 0\) (the concavity).
Axes: the parabola meets \(Oy\) exactly at \((0, c)\); it meets \(Ox\) in two, one, or zero points as \(\Delta > 0\), \(= 0\), \(< 0\) — and when \(\Delta = 0\) the touching point is the vertex itself.
Sketching: concavity, vertex, axis intersections, plus symmetry-mirrored points — five points and the direction suffice.
Shape: \(|a|\) alone sets the width (larger = narrower); \(b\) and \(c\) only move the vertex around the plane.
Rigidity: three points with distinct abscissas determine a quadratic function completely; two distinct parabolas share at most two points. And not every U-shaped curve is a parabola — the claim needs the function, not the impression.