Parabola Graph: y = x² and Its Family
Every quadratic equation graphs as a parabola: a symmetric curve with a single turning point. It is the first curve most students meet after the straight line, and the one that introduces the idea that an equation has a shape.
Open this graph in the calculator
What you are looking at
The teal curve is y = x², the simplest quadratic. It has its vertex at the origin, opens upward, and is symmetric about the y-axis. Every point on the right has a mirror image on the left at the same height.
The second curve, y = 0.5x² − 2, shows two changes at once. The coefficient 0.5 makes it wider than the first, and the −2 moves the whole curve down two units, putting its vertex at (0, −2).
The features to identify
Questions about parabolas almost always ask for the same handful of features, and all of them are visible on the graph.
- The vertex, which is the turning point: the minimum if the curve opens upward, the maximum if it opens downward.
- The axis of symmetry, the vertical line through the vertex.
- The y-intercept, where the curve crosses the vertical axis.
- The x-intercepts, also called roots or zeros, of which there may be two, one, or none.
- The direction of opening, decided purely by the sign of the squared term.
How the coefficients change the curve
Written as y = ax² + bx + c, each coefficient has a distinct job. Changing one at a time is the fastest way to see which.
| Change | Effect on the graph |
|---|---|
| a becomes larger | The curve narrows |
| a becomes smaller (towards zero) | The curve widens |
| a becomes negative | The curve flips to open downward |
| c changes | The whole curve moves up or down |
| b changes | The vertex slides diagonally, since it depends on both a and b |
Where parabolas come from
A thrown or launched object follows a parabolic path, because gravity applies a constant downward acceleration while horizontal speed stays roughly constant. That is why projectile problems are quadratic.
They also appear in optics and engineering. A parabolic mirror focuses every incoming parallel ray to one point, which is why satellite dishes, car headlights and reflecting telescopes all use the shape.
Questions
Why is it called a parabola?
From the Greek parabole, meaning comparison or application. It is one of the conic sections: slice a cone with a plane parallel to its side and the cut edge is a parabola.
Does every parabola have x-intercepts?
No. If the vertex sits above the x-axis on an upward-opening parabola, the curve never reaches it. Algebraically, that is a negative discriminant.
Is y = x³ a parabola?
No. That is a cubic, and it has a different shape: it passes through its centre rather than turning back, and it is not symmetric about a vertical line.