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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 $y=x_{2},$ 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 $y=x_{2}−4x,$ 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 $x=0.$

$x$ | $x_{2}−4x$ | $y$ |
---|---|---|

$-2$ | $(-2)_{2}−4(-2)$ | $12$ |

$-1$ | $(-1)_{2}−4(-1)$ | $5$ |

$0$ | $0_{2}−4(0)$ | $0$ |

$1$ | $1_{2}−4(1)$ | $-3$ |

$2$ | $2_{2}−4(2)$ | $-4$ |

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.

$x$ | $x_{2}−4x$ | $y$ |
---|---|---|

$3$ | $3_{2}−4(3)$ | $-3$ |

$4$ | $4_{2}−4(4)$ | $0$ |

$5$ | $5_{2}−4(5)$ | $5$ |

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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