1
Only a decides the direction
b and c slide the curve about; they never turn it over. y = x² - 100 opens upward and y = -x² + 100 opens downward, and the constants had nothing to do with it.
Lesson · Quadratics and Complex Numbers · Parabolas
Graphs of Quadratic Functions
The graph of y = ax² + bx + c is a parabola. The sign of a decides which way it opens, c is the y-intercept, the real solutions of ax² + bx + c = 0 are the x-intercepts, and the curve is symmetric about the line x = -b/(2a).
y = ax² + bx + c, a ≠ 0 the graph is a parabola
a > 0 opens upward a < 0 opens downward
y-intercept (0, c) x-intercepts: the real solutions
axis of symmetry x = -b/(2a), midway between the roots
Everything readable from a, b and c
Core ideas
What this lesson establishes
Four features, each one read from the equation rather than discovered by plotting.
2
Intercepts are the two solved problems you already have
Setting x = 0 gives the y-intercept, c, in one step. Setting y = 0 gives the x-intercepts, which is the equation the lessons from “Quadratic Equations and the Zero Product Property” to “The Discriminant” solved — so the discriminant counts them: two, one or none.
3
The curve is symmetric, exactly
Every parabola has a vertical mirror line, x = -b/(2a). Two points the same distance either side of it have the same height, which halves the work of plotting and checks the arithmetic for free.
The material
Reading a parabola off its equation
Direction, then intercepts, then the line the whole picture folds along.
Definition · Quadratic function
A function f(x) = ax² + bx + c with a, b, c real and a ≠ 0. The condition a ≠ 0 is not decoration: with a = 0 the graph is a straight line and none of this lesson applies.
The graph of a quadratic function is a curve called a parabola. Every quadratic gives one, and they are all the same shape up to stretching and moving — a fact “Vertex Form and the Axis of Symmetry” makes precise. This course meets the parabola only as the graph of a function; its geometric definition is a topic for conic sections and is not needed here.
The coefficient a does two jobs. Its sign sets the direction: a > 0 opens upward, a < 0 opens downward. Its size sets the steepness: at x = 2, y = x² is at height 4, y = 2x² at 8 and y = (1/2)x² at 2, so a larger |a| gives a narrower curve. The three parabolas are otherwise identical in shape.
The y-intercept is f(0) = c, and there is exactly one of them, because f is a function and Lines, Functions and Graphs fixed that a function assigns one output to the input 0. No vertical line can meet the graph twice, including the y-axis.
The x-intercepts are the values of x with f(x) = 0 — the real solutions of the quadratic equation. So the discriminant of “The Discriminant” counts them before any drawing: D > 0 gives two crossings, D = 0 gives one point where the curve touches the axis and turns back, and D < 0 gives none at all. A parabola with no x-intercept lies entirely on one side of the axis.
Theorem · Symmetry about x = -b/(2a)
Let f(x) = ax² + bx + c with a ≠ 0 and write h = -b/(2a). Then f(h + t) = f(h) + at² for every real t. In particular f(h + t) = f(h - t): the graph is symmetric about the vertical line x = h.
Expand: f(h + t) = a(h + t)² + b(h + t) + c = ah² + 2aht + at² + bh + bt + c.
Group the terms by their power of t: f(h + t) = (ah² + bh + c) + (2ah + b)t + at² = f(h) + (2ah + b)t + at².
Now 2ah + b = 2a·(-b/(2a)) + b = -b + b = 0, so the middle term vanishes and f(h + t) = f(h) + at².
Replacing t by -t leaves at² unchanged, so f(h - t) = f(h) + at² = f(h + t). Two points at equal distances either side of x = h have equal heights, which is what symmetry about that line means.
□
When there are two x-intercepts, the axis runs midway between them — the same conclusion reached from the formula rather than from the symmetry.
roots (-b + √D)/(2a) and (-b - √D)/(2a)
(-b + √D)/(2a) + (-b - √D)/(2a)
midpoint = ---------------------------------------
2
(-2b)/(2a) -b/a -b
= ---------- = ------ = ----
2 2 2a
Sketching from the intercepts
y = x² - 2x - 8 a = 1 > 0, so it opens upward
x -3 -2 -1 0 1 2 3 4 5
y 7 0 -5 -8 -9 -8 -5 0 7
|
axis x = 1
the table is symmetric about x = 1: y(-2) = y(4), y(0) = y(2)
Example · y = -x² + 4x - 3
a = -1, so the curve opens downward. f(0) = -3 gives the y-intercept (0, -3). Factoring, -x² + 4x - 3 = -(x - 1)(x - 3), so the x-intercepts are 1 and 3; the axis is x = 2, midway between them, and f(2) = 1. The highest point of the graph is (2, 1).
Sample points on both sides of the axis. A table for x = 0, 1, 2, 3 on a parabola whose axis is x = -4 returns four points from one arm, which join up into something that looks like a line bending — a picture that is not wrong so much as unrecognisable. Find the axis first, then choose the x-values.
When a quadratic factors, f(x) = a(x - r)(x - s) is worth writing down: the roots r and s are visible, the axis is at their midpoint (r + s)/2, and f(0) = a·r·s gives the y-intercept without expanding. “Vertex Form and the Axis of Symmetry” does the same job for the quadratics that do not factor.
Interactive
The parabola
Direction, intercepts, vertex, axis
Method
Sketching y = ax² + bx + c
Five numbers give the sketch. Four of them come from arithmetic you have already done.
1
Read the direction off a
Positive opens upward, negative downward. Do this first: it tells you which side of the axis the curve lives on if it has no x-intercepts.
2
Plot the y-intercept (0, c)
One substitution, x = 0, and it is free. Every parabola has exactly one.
3
Find the x-intercepts, or show there are none
Solve ax² + bx + c = 0 by whichever of “Solving by Factoring”, “The Square Root Property”, “Completing the Square” and “The Quadratic Formula” is quickest, or compute D and stop there if it is negative.
4
Draw the axis x = -b/(2a), then use symmetry
Plot the point on the axis, (h, f(h)), and reflect every other point you have across it. Two or three plotted points and the mirror line are enough for an honest sketch.
Reading the graph
Score 0 / 0Choose the answer you can justify.
Sketching y = x² - 2x - 8
a = 1, b = -2, c = -8. Nothing is plotted until the four features are written down.
direction a = 1 > 0 opens upward
y-intercept f(0) = -8 (0, -8)
x-intercepts x² - 2x - 8 = (x - 4)(x + 2)
x = 4 and x = -2
axis x = -b/(2a) = 2/2 = 1 midpoint of -2 and 4
point on it f(1) = 1 - 2 - 8 = -9 (1, -9)
symmetry check
f(-2) = 0 = f(4) both 3 units from x = 1 ✓
f( 0) = -8 = f(2) both 1 unit from x = 1 ✓
Five numbers determine the sketch: the direction, the y-intercept, two x-intercepts and the point on the axis. Everything else in the drawing is symmetry, not information.
The point (1, -9) is the lowest point of this curve. It is called the vertex, and “Vertex Form and the Axis of Symmetry” gets it straight from the equation without solving anything or plotting anything.
For a faded sketch, analyse y = -2x² - 4x + 6. The supplied first decision is a = -2 < 0, so the curve opens downward. Find both intercepts, the axis and the point on it before drawing; then use a symmetric pair of values as a check. Compare with x-intercepts -3 and 1, axis x = -1, vertex (-1, 8) and y-intercept (0, 6).
Common mistakes
1
Deciding the direction from c, or from the rootsThe constant term moves the curve up and down and the roots say where it crosses. Neither turns it over: y = x² - 100 opens upward even though c is very negative.
2
Sampling x-values on one side of the axisFour points from one arm look like a bending line. Compute h = -b/(2a) first and take values either side of it — and the pairs you get should have equal heights, which is a free check on the arithmetic.
3
Calling a parabola with no x-intercept a broken graphy = x² + 1 is a perfectly ordinary function with a value at every real x. It never takes the value 0, and no rule requires a function to. An x-intercept is where the output is zero, not a feature every curve must have.
Check your understanding
✓
Finish when you can sketch without a table of values.
Given any quadratic you should produce the direction, the y-intercept, the x-intercepts or a reason there are none, and the axis — from the coefficients alone. Plotted points should confirm the sketch, never be the way you discover it.
Note: The lowest or highest point has been treated so far as "the point on the axis", computed by substituting h into f. “Vertex Form and the Axis of Symmetry” gives it a name and a form of the equation that displays it, and “Maximum and Minimum Problems” turns that into the tool for optimisation problems.