GQF - Characteristics of Quadratic Functions (Lesson)

Characteristics of Quadratic Functions

Interval notation uses many of the same properties as inequality notation. Check out this table to see the similarities and differences:

Phrase	Inequality Notation	Interval Notation	Notes
All values greater than 5	x > 5	$LaTeX: x\in\left(5,\infty\right)$ $x\in\left(5,\infty\right)$	When you would have used a greater than symbol, you should now use a parenthesis. The " $LaTeX: x\in$ $x\in$ " means x is an element of, or x belongs to this set.
All values between -2 and 17, inclusive	$LaTeX: -2\le x\le17$ $-2\le x\le17$	$LaTeX: x\in\left[2,\:17\right]$ $x\in\left[2,\:17\right]$	When you would have used a less than or equal to, you use a bracket.
All values less than or equal to 8	$LaTeX: x\le8$ $x\le8$	x $LaTeX: \in$ $\in$ (- $LaTeX: \infty$ $\infty$ ,8]	Again, we use a bracket for the less than or equal to. Also notice that we do not put a bracket around infinity! Infinity can never have a bracket around it.

Inequality Notation to Interval Notation Practice

Try these problems to see if you've got it. Convert each statement from inequality notation to interval notation.

$LaTeX: x\ge-9$ $x\ge-9$
$-2<x<12$
$x<-2$
$LaTeX: x\le-2$ $x\le-2$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Now that you know how to graph quadratic functions in both forms, it is important you know how to pick out some additional important characteristics! So let's start by defining those:

Minimum Value - The lowest y-value a parabola reaches		When a parabola opens up. The parabola has a minimum value and it is always y_v
Maximum Value - The highest y-value a parabola reaches.		When a parabola opens down. The parabola has a maximum value and it is always y_v
Domain - The set of all possible values to input for x.		For all quadratic functions, the domain is all real numbers because you are allowed to put any value in for x. To represent all real numbers, we say: $LaTeX: x\in\left(-\infty,\infty\right)$ $x\in\left(-\infty,\infty\right)$
Range - The set of all possible output values.		The range would be all of the y-values above the minimum OR below the maximum, depending on what direction the parabola opens.
Intervals of increase or decrease - The set of all x-values for which the function is increasing or decreasing.

Watch this video to practice identifying additional key features of quadratic functions:

Characteristics of Quadratic Functions Practice

Try these problems. Find the vertex, axis of symmetry, max or min and the values, domain, range, intervals of increase or decrease.

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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