Interactive Parabola Grapher: Vertex, Standard and Factored Form

The three forms are one curve#Link to the section: The three forms are one curve

f(x) = a(x − h)² + k, f(x) = ax² + bx + c and f(x) = a(x − r₁)(x − r₂) are not three kinds of parabola. They are one parabola written three ways, and the tool above draws all three at once so you can watch them stay on top of each other while you drag.

The reason to know all three is that each one gives you a fact without any work:

• Vertex form hands you the vertex: it is (h, k), read straight off.
• Factored form hands you the roots: they are r₁ and r₂, read straight off.
• Standard form hands you the y-intercept: it is c, read straight off.

A Digital SAT question that asks for the minimum value of a quadratic is asking for k. One that asks where the graph crosses the x-axis is asking for r₁ and r₂. Recognising which form you have been given, and which one the question wants, is most of the work.

What a, h and k each do#Link to the section: What a, h and k each do

The coefficient a controls the width and the direction. Positive a opens upwards, negative opens downwards, and larger |a| makes the curve narrower. It never moves the vertex.

The value h slides the curve left and right — and it slides it the opposite way to its sign, which is the single most common slip on this topic. f(x) = (x − 3)² has its vertex at x = 3, not −3. The sign inside the bracket is subtracted from x, so the curve moves to where that subtraction gives zero.

The value k slides the curve up and down, straightforwardly. The vertex is at height k.

Set a = 1, h = 0, k = 0 in the tool and change one slider at a time. Watching h alone move the curve right when the number in the bracket goes up is worth more than reading the rule again.

Two roots, one root, or none#Link to the section: Two roots, one root, or none

The vertex is the closest the curve ever gets to the x-axis on its own side. So:

• If the vertex is below the axis and the curve opens upwards, it crosses twice.
• If the vertex is exactly on the axis, it touches once — a repeated root.
• If the vertex is above the axis and the curve opens upwards, it never crosses, and the roots are not real numbers.

In vertex form that is the sign of −k/a: the roots are h ± √(−k/a), which is a real number exactly when −k/a is not negative. That is the same information the discriminant b² − 4ac carries in standard form; vertex form just makes it visible.

The tool prints roots exactly. Set a = 1, h = 0, k = −2 and the roots read ±√2 rather than ±1.41, because the SAT grid-in wants the exact value.

Completing the square, which is the move between the forms#Link to the section: Completing the square, which is the move between the forms

Going from factored or vertex form to standard form is expanding, and it is mechanical. Going the other way — standard to vertex — is completing the square, and it is the step most people skip until a question forces it.

Worked example. Take y = 2x² − 12x + 13. Take the 2 out of the first two terms: y = 2(x² − 6x) + 13. Half of −6 is −3, and (−3)² is 9, so x² − 6x = (x − 3)² − 9. Substituting back: y = 2((x − 3)² − 9) + 13 = 2(x − 3)² − 18 + 13 = 2(x − 3)² − 5.

So the vertex is (3, −5), which the standard form was carrying the whole time and would not tell you. Set a = 2, h = 3 and k = −5 in the tool above and the standard-form line reads y = 2x² − 12x + 13 back at you.

How to use the grapher#Link to the section: How to use the grapher

The three sliders are a, h and k. Every readout updates on every step, so the fastest way to learn the shape of this is to hold two still and sweep the third.

The curve legend lets you hide any of the three forms, which is worth doing once: hide vertex and factored form, look at the standard form alone, and try to say where the vertex is before you switch them back on.

Scroll or pinch to zoom, drag the background to pan, and press the house button to reset the window. Tapping the curve marks the coordinates at that point.