Quadratic Functions and Equations
What a quadratic function looks like
A monic quadratic function is any rule of the form y = x² + bx + c, and its graph is always a smooth U-shaped curve called a parabola. The squared term is what bends the line into a curve: as you move away from the centre in either direction, the output grows ever faster. This unit is about reading that curve, drawing it from a function, and solving the equation that asks where the curve meets the x-axis, using three complementary tools.
The basic parabola from a table
The cleanest way to meet a parabola is to build a table of values for the simplest case, y = x². Choosing x from −3 to 3 gives outputs 9, 4, 1, 0, 1, 4, 9. Two features jump out immediately. The curve is symmetric: the value at −2 equals the value at 2, because squaring erases the sign. And it has a lowest point, the vertex, sitting at the origin, from which both arms sweep upward. Because the coefficient of x² is positive, the parabola opens upward and the vertex is a minimum.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
Moving the parabola: axis and vertex
Shifting to a general monic quadratic keeps the same shape but moves it around the plane. For y = x² − 2x − 3, a table from −2 to 4 gives 5, 0, −3, −4, −3, 0, 5. The same mirror symmetry appears, this time about the vertical line x = 1, which is the axis of symmetry. The vertex sits on that axis at the point (1, −4), the lowest point of this particular curve. Plotting these points and joining them with a smooth curve, never a series of straight segments, produces the graph.
Solving by reading the graph
Solving a quadratic equation means finding the x-values where y is zero, which graphically are the points where the parabola crosses the x-axis. For y = x² − 2x − 3, the curve cuts the axis at x = −1 and x = 3; these crossing points are the solutions, also called the roots. Reading solutions straight off a graph is the graphical method, and it works for any quadratic, though it gives only as much precision as the graph allows.
Solving monic equations by factorising
When a quadratic is monic and its roots are whole numbers, there is a fast exact method: factorising. To solve x² − 5x + 6 = 0, factorise the left side into (x − 2)(x − 3). A product is zero only when one of its factors is zero, so x = 2 or x = 3. These match the points where the corresponding parabola would cross the axis. Notice the roots are 2 and 3, not −2 and −3; the factors x − 2 and x − 3 are zero at positive values. This algebraic route is reserved for monic quadratics with integer roots, where the factors are easy to find.
When roots are not whole numbers
Not every quadratic cooperates so neatly. For x² − 2x − 1 = 0, no two whole numbers multiply to −1 and add to −2, so factorising into integers is impossible. You fall back on the graphical or numerical methods: the curve crosses the x-axis between −1 and 0 and again between 2 and 3, and a table of values or graphing software homes in on about −0.41 and 2.41. These roots are irrational, exactly 1 − √2 and 1 + √2, never a terminating decimal. Graphing software is invaluable for exactly these cases, and for quickly checking any solution you found by hand. An equation such as x² = 5 is different: taking the square root of both sides gives the exact roots ±√5 straight away.
Two cautions: no crossings, and which way it opens
Two cautions round out the picture. First, a parabola need not cross the x-axis at all; y = x² + 1 sits entirely above the axis, so the equation x² + 1 = 0 has no real solutions, and its graph confirms this at a glance. Second, always confirm whether the curve opens up or down before trusting a vertex to be a minimum or a maximum. For the monic functions in this unit the x² coefficient is positive, so every parabola opens upward and every vertex is a lowest point.