# Quadratic graphs

The graph of an equation with x^2 in it is a smooth U-shaped curve called a parabola.

Maths (school) · Part 04: Graphs · Card 6 of 8 · Last updated 2026-10-19

## Analogy

Like the path of a ball thrown into the air and falling back down.

## Example

**y = x^2 - 4x + 3**

Crosses x-axis at x = 1 and x = 3
Turning point at x = 2: y = 4 - 8 + 3 = -1
Lowest point is (2, -1)

## In practice

Bridge arches, satellite dishes and the arc of a basketball shot are all quadratic shapes.

## Practice

### Flashcards

- **What shape is the graph of y = x^2 - 4x + 3?** A U-shaped curve (parabola) with its lowest point at (2, -1).

### Interview questions

**Describe the graph of y = x^2 - 4x + 3 and give its turning point.**

- It is a U-shaped parabola
- It crosses the x-axis at x = 1 and x = 3
- The turning point is halfway, at x = 2
- y = -1 there, so it is (2, -1)

## Questions people ask

### What is the turning point of a quadratic graph?

The turning point is where the curve changes direction. It is the lowest point of a U-shape or the highest point of an upside-down U. It lies halfway between the two places where the curve crosses the x-axis, if it does.

### What decides if the parabola is a U or an upside-down U?

The sign of the x^2 term. A positive x^2, as in y = x^2 - 4x + 3, gives a U shape that opens upwards. A negative x^2, as in y = -x^2 + 4, gives an upside-down U that opens downwards.

## Related concepts

- [Coordinates](https://studyvisually.com/learn/science/maths-school/maths-school-part-4/maths-school-part-4-concept-1)
- [Gradient](https://studyvisually.com/learn/science/maths-school/maths-school-part-4/maths-school-part-4-concept-2)
- [y = mx + c](https://studyvisually.com/learn/science/maths-school/maths-school-part-4/maths-school-part-4-concept-3)
- [Parallel and perpendicular lines](https://studyvisually.com/learn/science/maths-school/maths-school-part-4/maths-school-part-4-concept-4)

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