Graphing Quadratics: Roots, Vertex and Where Two Curves Cross

Plotting a curve is the easy part. Getting a specific number off the plot, and knowing whether it is exact, is where the time goes.

Roots are where the curve meets the x-axis

Take y = x^2 - 6x + 8. Plot it and the parabola cuts the axis twice. The calculator reports x = 2 and x = 4, and you can check both by hand: 4 - 12 + 8 = 0, and 16 - 24 + 8 = 0. It factors as (x - 2)(x - 4).

Reading roots off a graph is fast but approximate. If the calculator hands you 1.99997, the real answer is almost certainly 2 and the cursor is quantised to pixels. Use the root or zero function when you need more than one decimal place.

When the curve never touches the axis

Try y = x^2 + x + 3. The parabola floats above the x-axis and there are no roots to find. The discriminant explains it: b^2 - 4ac is 1 - 12, or -11. A negative discriminant means two complex roots and no crossing point on a real plot. A discriminant of exactly zero means the curve touches the axis once, at the vertex. Checking it first saves a lot of scrolling. The quadratic formula walkthrough covers the algebra behind that.

The vertex is the maximum or the minimum

For any quadratic in the form y = ax^2 + bx + c, the vertex sits at x = -b / 2a. In the first example that is 6 / 2, so x = 3. Substitute back: 9 - 18 + 8 = -1. The vertex is (3, -1), and because a is positive it opens upward, so that is a minimum.

This is the part with use outside a maths class. If the equation models profit against price, the vertex is the price that maximises profit. If it models a thrown ball, the vertex is the top of the arc.

Two curves crossing is a system of equations

Plot y = 2x + 1 and y = x^2 - 2 on the same axes and they meet twice. Set them equal to find where: x^2 - 2 = 2x + 1, so x^2 - 2x - 3 = 0, which factors to (x - 3)(x + 1) = 0. The solutions are x = 3 and x = -1. Put those back into the straight line: at x = 3, y = 7; at x = -1, y = -1. The crossings are (3, 7) and (-1, -1).

The graph and the algebra should agree exactly. When they do not, one has a typo, usually the sign in front of a coefficient.

Getting a usable plot on screen

Enter each equation on its own line as y = something, using ^ for powers. Implicit multiplication such as 2x is accepted by most tools, but 2(x+1) is safer written as 2*(x+1).

If the screen looks blank, fix the window

A blank plot almost always means the curve is outside the viewing window, not that the equation is broken. Default windows are often -10 to 10 on both axes. A parabola with its vertex at (3, -1) fits fine; one at (0, 240) does not. Widen the y range first; curves grow much faster vertically than horizontally. For what the different free tools give you here, see the roundup of free graphing calculators.

Common questions

How do I find where a curve crosses the y-axis? Set x = 0 and read the constant term. For y = x^2 - 6x + 8 that is 8, so the y-intercept is (0, 8).

Why do my roots come out as long decimals? Because most quadratics do not factor neatly. If the discriminant is not a perfect square, the roots are irrational and a decimal is the honest answer. Keep the surd form if the question asks for an exact value.

Can a graphing calculator solve for x directly? Most will, through a solve or zero function, but plotting first is worth ten seconds. The shape tells you how many solutions to expect, which catches the case where a solver returns one root and ignores the other.

What does a cubic look like compared with a quadratic? A quadratic has one turning point. A cubic has up to two, and it always crosses the x-axis at least once.