# Quadratic Functions

So, you already know about functions. They sit there, looking all *y* = *f*(*x*), showing a relationship between the variables *x* and *y*. We also have this quadratic equation just sitting here, not doing anything. How about we mash these two concepts together, all Frankenstein-style?

With a dramatic crash of thunder, we have:

*f*(*x*) = *ax*^{2} + *bx* + *c*

A quadratic function. Dun dun dunnnn.

What do we do with functions? We graph them, of course.

### Sample Problem

Graph the quadratic function *f*(*x*) = *x*^{2}.

We start off by finding a few ordered pairs and putting them in a table.

x | f(x) |

0 | 0 |

1 | 1 |

-1 | 1 |

2 | 4 |

-2 | 4 |

3 | 9 |

-3 | 9 |

By finding even more points, we could show that the graph will continue to curve upwards, reaching larger and larger values of *y* for the next value of *x*. In fact, we'll show that with our next…

### Sample Problem

Graph the quadratic function *f*(*x*) = -2*x*^{2} + *x* + 10.

Again we start off by finding a few points to get an idea about the general shape of the graph.

x | f(x) |

0 | 10 |

1 | 9 |

-1 | 7 |

2 | 4 |

-2 | 0 |

3 | -5 |

-3 | -11 |

Now this graph looks similar to the previous one, but it's curved down. Both of these graphs, and all others made by quadratic functions, are called **parabolas**. In addition to their good looks, all parabolas share several other common features.

- They have a single maximum or minimum point, called the
**vertex**. The first graph above has a minimum and always increases in both directions away from the vertex, while the second graph has a maximum and is always decreasing.

- Parabolas also have good posture, so they're always symmetric across some vertical line
*x*=*h*, with the axis of symmetry passing through the vertex (*h*,*k*).

- All parabolas have one
*y*-intercept, but parabolas can have 0, 1, or 2*x*-intercepts.

### Sample Problem

Graph the quadratic function *f*(*x*) = -*x*^{2} + 8*x* – 10.

Another day, another table.

x | f(x) |

0 | -10 |

1 | -3 |

-1 | -19 |

2 | 2 |

-2 | -30 |

3 | 5 |

-3 | -43 |

That…doesn't look too good. This is a quadratic function, so we know that the graph should look like a parabola. We must be missing some crucial points. As *x* becomes more negative, the values of *y* become lower and lower, so the vertex must be to the right of what we have already. Some experimentation finds:

x | f(x) |

4 | 6 |

5 | 5 |

6 | 2 |

That looks better. In the next few sections, we'll learn some handy tricks for graphing parabolas that'll make things loads easier.