Section 1.2 Describing Function Behavior
ΒΆObjectives: Student Learning Outcomes
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.
Subsection 1.2.1 Intervals
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 \((a,b)\text{.}\)
Practice writing inequalities to describe the shaded intervals on each number line.
Checkpoint 1.2.1.
Subsection 1.2.2 Function Characteristics
A function can be positive or negative or zero. Really we mean the outputs of the function are positive or negative or \(0\text{.}\) If we say \(y = f(x)\) then positive means the function produces y-values that are greater than zero while negative means the function produces y-values that are less than zero. When a function is zero it means the y-value came out to be zero.
On a graph, 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. If the output value is zero, touching the horizontal axis, we say the function is zero (neither positive nor negative).
Checkpoint 1.2.2. Positive or Negative.
A function can be increasing or decreasing or constant.
If the output values of the function increase as the input increases, we say the function is increasing. If the output values of the function decrease as the input increases, we say the function is decreasing. Constant means the function keeps the same y-value as the x-values get larger.
Another way to think about it is to first assume the inputs (x-values for a function) get larger. If the y-values also get larger then the function is increasing. If the y-values get smaller then the function is decreasing.
When we express whether a function is increasing, decreasing or constant on a closed interval we always include the endpoints of the interval. For example we could say \(f(x)\) is increasing on the interval \(-3\leq x\leq 5\text{.}\) Using interval notation we could say the same thing as \((-\infty, -3]\) or \([5, \infty)\text{.}\) In this notation notice we use parentheses at infinity because infinity is not a number, so it is not possible to use it in a function.
Reading the graph from left-to-right, a function is increasing if its graph goes up and decreasing if its graph goes down.
Checkpoint 1.2.3. Increasing or Decreasing.
A function can be concave up or concave down.
If a function curves upward (like a cup that holds water), we say the function is concave up. If a function curves downward (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.
Remember to include the end values of an interval when describing the concavity of a function.
Checkpoint 1.2.4. Concave Up or 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.
Checkpoint 1.2.5.
Finally, let's put these characteristics into context. In the next problem, you will describe the characteristics of various functions given by verbal descriptions.