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Like linear and exponential functions, quadratic functions are a unique type of functions that have specific qualities in common. Analyzing these functions in terms of their *characteristics* allows important information to be learned.

A quadratic function is a function of degree 2. That means that the highest exponent of the independent variable is 2. The simplest quadratic function is y=x2, and the graph of any quadratic function is a parabola.

The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
### Concept

### Direction

A parabola either opens **upward** or **downward**. This is called its direction.
### Concept

### Vertex

Because a parabola either opens upward or downward, there is always one point that is the absolute maximum or absolute minimum of the function. This point is called the vertex.
### Concept

### Axis of Symmetry

All parabolas are symmetric, meaning there exists a line that divides the graph into two mirror images. For quadratic functions, that line is always parallel to the y-axis, and is called the axis of symmetry.

### Concept

### Zeros

Depending on its rule, a parabola can intersect the x-axis at 0, 1, or 2 points. Since the function's value at an x-intercept is always 0, these points are called zeros, or sometimes roots.
### Concept

### y-intercept

Because all graphs of quadratic functions extend infinitely to the left and right, they each have a y-intercept anywhere along the y-axis.

At the vertex, the function changes from increasing to decreasing, or vice versa.

The axis of symmetry always intersects the vertex of the parabola, and is written as a vertical line, where h can be any real number.

x=h

For the quadratic function

y=x2−4x,

create a table of values to graph it. Then determine its direction, vertex, zeros, and axis of symmetry. Show Solution

To begin, we'll use the function rule to create a table of values. Then, to graph the function, we'll plot the points from the table. Let's start with x=0.
Thus, the point (0,0) lies on the parabola. We can perform the same calculations for other x-values around x=0.

x | x2−4x | y |
---|---|---|

$-2$ | $(-2)_{2}−4(-2)$ | 12 |

$-1$ | $(-1)_{2}−4(-1)$ | 5 |

0 | 02−4(0) | 0 |

1 | 12−4(1) | -3 |

2 | 22−4(2) | -4 |

We'll plot these points on a coordinate plane.

We can start to see the left-hand side of the parabola. Let's add a few more x-values to the table to determine a more complete shape.

x | x2−4x | y |
---|---|---|

3 | 32−4(3) | -3 |

4 | 42−4(4) | 0 |

5 | 52−4(5) | 5 |

We'll add these points to the coordinate system as well.

Looking at the points, we now see both sides of the parabola. We can connect the points with a smooth curve.

The graph can be used to describe the desired characteristics of the parabola.

$directionaxis of symmetryvertexzeros :upward:x=2:(2,-4):(0,0)and(4,0) $

Three quadratic functions are graphed in the coordinate plane.

For each graph, match it with the corresponding characteristics.$direction:vertex:vertex:axis of symmetry:y-intercept:zero: upward,downwardminimum,maximum(-2,2),(0,-6),(2,-4)x=-2,x=2,x=0y=-6,y=-2,y=0,y=4x=0,x=4,x=-6 $

Show Solution

Instead of looking at each function separately, we'll look at the characteristics individually and summarize our findings in a table at the end.

Function | A | B | C |
---|---|---|---|

direction | upward | downward | upward |

max/min | minimum | maximum | minimum |

vertex | (2,-4) | (-2,2) | (0,-6) |

axis of symmetry | x=2 | x=-2 | x=0 |

y-intercept | y=0 | y=-2 | y=-6 |

zeros | x=0 and x=4 | not applicable | not applicable |

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