# Increasing and Decreasing Functions

## x value intervals where the y values are moving up or down.

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Increasing and Decreasing Functions

How do you describe the behavior of a function? One useful way is to identify it as increasing or decreasing, meaning the graph goes up or down from left to right. What about graphs that are not straight lines? What if they increase and decrease in different places on the same graph?

### Increasing and Decreasing Functions

In this lesson we will consider the function values in between extrema (extremes: minimums or maximums), focusing on where function values are increasing or decreasing.

#### Formally

A function is increasing on some interval of its domain if f(a) > f(b) for all a, b in that interval such that a > b.

A function is decreasing on some interval of its domain if f(a) < f(b) for all a, b in that interval such that a > b.

#### Informally

A function is increasing on a section if the graph of that section 'rises' to the right.

A function is decreasing on a section if the graph of that section 'falls' to the right.

### Examples

#### Example 1

Earlier, you were given a question about describing the behavior of a function.

Graphs with curves or which go up and down throughout the graph may be described with the use of intervals. By splitting the description of the graph into sections, each with consistent behavior throughout the section, the entirety of the graph may be evaluated.

#### Example 2

Describe the behavior of the graph below, which shows the federal minimum wage from 1975 to 2005:

Overall, from 1975 to 2005, the minimum wage (in dollars per hour) increased from $2.10 per hour to$5.15 per hour. However, if we examine the graph on smaller intervals, we can see that this increase was not steady. For example, the minimum wage increased from $2.30 in 1977 (year 3) to$3.35 in 1981 (year 7), and then it was constant for several years at 3.35 per hour. We can describe this situation using intervals. For example, we can say that the function increases on the interval (3,7) and is constant on the interval (7, 15). #### Example 3 The graph below shows the minimum wage data adjusted for inflation. (Since prices for food and other necessities tend to go up over time, comparing absolute dollar amounts doesn’t make sense. The adjustment shows how much one can buy with minimum wage.) State the intervals on which the graph increases and the intervals on which it decreases. Use proper interval notation. The function decreases on the intervals (2, 3), (5, 15), (18, 21), (24, 31). The function increases on the intervals (1, 2), (3, 4), (15, 17), (22, 24). Notice that this is a discrete function, and the intervals are ranges of specific points. That is, the function is decreasing if, when we look at the graph, the points are sloping down from left to right. The function is increasing if the points are sloping up from left to right. This will be the case for any discrete function. #### Example 4 Identify the intervals of increase and decrease, use interval notation: \begin{align*}y = x^{3} - 3x\end{align*} The function increases on the interval \begin{align*}(- \infty, -1)\end{align*} and on the interval \begin{align*}(1, \infty)\end{align*}. The function decreases on the interval \begin{align*}(-1,1)\end{align*}. These are open intervals (with parentheses instead of brackets) is because the function is neither increasing nor decreasing at the moment it changes direction. We can imagine a ball thrown into the air. The height increases up to a maximum point before it starts to decrease. What happens at that maximum point? There is an instant when the height is not changing; it is neither increasing nor decreasing, so those specific values are not included on the interval. #### Example 5 Give an example of a discrete function that increases and decreases on different intervals of its domain. Answers will vary, but should describe a function such as a person's height compared to his or her age. That is, values will increase until the age at which the person stops growing. Then the function will be constant for several decades. Then the function will decrease slightly, as some people lose height in their later years. #### Example 6 Kim earns tips at work on Mon, Wed, and Fri. She likes to go to the movies with her friends on Tues. evenings when there are no crowds. On Saturdays, she and her boyfriend go skating in the park and always stop for ice cream. Describe the intervals where Kim's money is increasing and decreasing. Kim's money is increasing for the interval between Monday and Tuesday evening, decreasing for the Tuesday to Wednesday interval, increasing during the interval between work on Wednesday and ice cream on Saturday, and finally, decreasing between ice cream on Saturday and work on Monday. ### Review For questions 1 and 2, consider the following graph: 1. Approximate the coordinates of the relative maximum of the graph 2. Approximate the coordinates of the relative minimum of the graph. 3. Consider the equation of a linear function y = mx, where m is the slope of the line. For what values of m will the function be an increasing function? For what values of m will the function be a decreasing function? 4. Consider the graph above for questions 2 and 3. On what intervals is the function increasing? On what intervals is the function decreasing? 5. Sketch a possible graph of the function described here: The function is continuous on \begin{align*}\mathbb R\end{align*}. It is decreasing on the interval \begin{align*}(-\infty, 2)\end{align*} and increasing on the interval \begin{align*}(2, \infty)\end{align*}. 6. Explain in your own words the difference between relative and global extrema. 7. For the function shown in the graph below, give approximate coordinates of all global and relative extrema. 8. Consider the function \begin{align*}f(x)=x^2+\frac{2}{x}\end{align*}. Use a graphing utility to sketch a graph of this function and to calculate a relative minimum. 9. A rectangle is inscribed in a semi-circle of radius 3. a) Write an equation to represent the area of the rectangle as a function of x. b) Graph the equation and calculate the maximum area of the rectangle. 10. How many relative extrema should there be on a graph of \begin{align*}y = x^6\end{align*}? 11. Sketch a graph of a function which has an absolute maximum, but no local maximum, on the interval [-5, 7]. 12. Sketch a graph of a function with two local minimums on the interval (-4, 9], and no global extrema. For questions 13 - 15, use the following information: Sam makes7.50 per hour at work, and he works anywhere from 15 - 40hrs per week. Occasionally the store gets really busy, and Sam's boss allows him to work up to 15hrs of overtime. Sam loves the busy weeks, because he makes "time-and-a-half", or \$11.25, for overtime hours.

1. Sketch a graph showing Sam's regular income range. Where are the local extrema?
2. What happens to the local extrema if Sam is working a week with overtime? Where are they now?
3. Sketch another graph including Sam's regular and overtime income, identifying all local extrema.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.7.

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### Vocabulary Language: English

TermDefinition
Continuous Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.
decreasing A function is decreasing over an interval if its $y$ values are getting smaller over the interval. The graph will go down from left to right over the interval.
decreasing function A decreasing function is one with a graph that goes down from left to right.
Function A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Global Maximum The global maximum of a function is the largest value of the entire function. Symbolically, it is the highest point on the entire graph.
Global Minimum The global minimum of a function is the smallest value of the entire function. Symbolically, it is the lowest point on the entire graph.
increasing A function is increasing over an interval if its $y$ values are getting larger over the interval. The graph will go up from left to right over the interval.
increasing function An increasing function is one with a graph that goes up from left to right.
interval An interval is a specific and limited part of a function.
Interval Notation Interval notation is the notation $[a, b)$, where a function is defined between $a$ and $b$. Use ( or ) to indicate that the end value is not included and [ or ] to indicate that the end value is included. Never use [ or ] with infinity or negative infinity.
monotonic A function is monotonic if it does not switch between increasing and decreasing at any point.
relative extrema The relative extrema of a function are the points of the function with $y$ values that are the highest or lowest of a local neighborhood of the function.
strictly Strictly is an adjective that alters increasing and decreasing to exclude any flatness or periods where $y$ values are staying constant.

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