# The graph of y = ax^2 + bx + c

A nonlinear function that can be written on the standard form

$$ax^{2}+bx+c,\: \: where\: \: a\neq 0$$

is called a quadratic function.

All quadratic functions has a U-shaped graph called a parabola. The parent quadratic function is

$$y=x^{2}$$

The lowest or the highest point on a parabola is called the vertex. The vertex has the x-coordinate

$$x=-\frac{b}{2a}$$

The y-coordinate of the vertex is the maximum or minimum value of the function.

a > 0 parabola opens up minimum value

a < 0 parabola opens down maximum value

A rule of thumb reminds us that when we have a positive symbol before *x ^{2}* we get a happy expression on the graph and a negative symbol renders a sad expression.

The vertical line that passes through the vertex and divides the parabola in two is called the axis of symmetry. The axis of symmetry has the equation

$$x=-\frac{b}{2a}$$

The y-intercept of the equation is c.

When you want to graph a quadratic function you begin by making a table of values for some values of your function and then plot those values in a coordinate plane and draw a smooth curve through the points.

**Example**

Graph

$$x=x^{2}+2x+1$$

Make a table of value for some values of x. Use both positive and negative values!

x |
y = x^{2} + 2x + 1 |

-3 | 4 |

-2 | 1 |

-1 | 0 |

0 | 1 |

1 | 4 |

2 | 9 |

3 | 16 |

Graph the points and draw a smooth line through the points and extend it in both directions

Notice that we have a minimum point which was indicated by a positive a value (a = 1). The vertex has the coordinates (-1, 0) which is what you will get if you use the formula for the x-coordinate of the vertex

$$x=-\frac{b}{2a}=-\frac{2}{2\cdot 1}=-1$$

and that the line has an y-intercept of (0, 1) which could have been determined from the c-value which is 1.

If you have an absolute value of a that is greater than 1 the parabola will be narrower than the parental quadratic function. And the opposite that if you have a absolute value of a that is less than 1 then the parabola will be wider than the parental quadratic function.

Here you can get a visual of your quadratic equations

**Video lesson**

Graph the function

$$x^{2} - 3x - 10$$