**What is a Quadratic Relation?**

A quadratic relation is a non-linear relation, in which the *second differences *are constant (and the first differences are not). When the coordinates of a quadratic relation are plotted onto a graph, they can be connected to form a *curved*, U-shaped line called a *parabola.*

**Second Differences!**

To better understand the concept of quadratic relations, let’s look at an example.

Here we have a quadratic relation, set up in our standard table of values:

x | y |

1 | 0 |

2 | 2 |

3 | 6 |

4 | 12 |

5 | 20 |

You may be wondering, *how would I know if this was a quadratic relation or not?*

- We can identify if a relation is quadratic or not by examining second differences.

- First, we must find the
*first**differences –*Remember, these are the differences between each of the y-values. We subtract a y-value by the value listed on the row above it to find its first difference (*this is how we find the first differences if the x-values increase/decrease at a constant rate on the table of values).

- Now that we have found our first differences, we can find the second differences. Finding the second difference is like finding the first difference. We subtract a first difference value by the number listed on the row above it.

Now, notice how the second differences are *constant, *and the first differences are not. This means that the relation showed in the table is quadratic!

(Remember: in a linear relation, the first differences are constant.)

**Parabolas**

In a linear relation, the coordinates can be connected to form a line.

In a quadratic relation, the coordinates can be connected to form a curved, U-shaped line called a parabola.

**Parabolas and Second Differences**

When the second differences are positive, the parabola opens upward.

When the second differences are negative, the parabola opens downward.

(When the second differences are 0, the relation is linear and there is no parabola because the first differences are equal.)

This is the first step in understanding quadratic relations! Stay posted for more articles on quadratics and anything math-related!

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