After completing this lesson you will be able to:
Define an interval and use inequalities to describe it.
Identify intervals on which a function is increasing or decreasing, positive or negative, concave up or concave down.
In this activity we use function notation to describe the characteristics and behavior of a function.
The prerequisites for this lesson are knowing how to read and write inequalities.
An interval is a section or portion of either the vertical or horizontal axis. In this course we will use inequalities to describe intervals, although you may have already experienced other notations such as brackets \([a,b]\) or parentheses \((a,b)\text{.}\)
In the execise below, practice writing inequalites to describe the shaded intervals on each number line.
A function can be Positive or Negative.
If the output values of a function are above the horizontal axis, we say the function is positive. If the output values of a function are below the horizontal axis, we say the function is negative.
A function can be Increasing or Decreasing.
If the values of the function (output) increase as the input increases, we say the funciton is increasing. If the values of the function decrease as the input increases, we say the function is decreasing.
A function can be Concave Up or Concave Down.
If a function bends upwards (like a cup that holds water), we say the function is concave up. If a function bends downwards (like an inverted cup that does not hold water), we say the function is concave down.
Another way to think about concavity is to imagine a straight metal wire. While one end of the wire is fixed, if the other end is pushed up the wire is now concave up. If that other end is pushed down the wire is concave down.
Now let's put all these function characteristics on the same graph. Using 24 hour time with midnight at \(t = 0\text{,}\) the graph in the next exercise shows the temperature variation in a small northern town during one day.