Explore: Increasing and Decreasing Intervals with Technology Solutions

Increasing and Decreasing Intervals with Technology Solutions

Because turning points are neither increasing nor decreasing, they can be used to define intervals across the x-axis.

How to Describe Intervals Using Turning Points (and End Behaviors)

Label turning points on the graph. $(x_{1}, y_{1}), (x_{2}, y_{2})$
Moving left to right, define intervals (in interval notation) using turning points and end behaviors, such as:
- From an end behavior to a turning point $(- \infty, x_{1})$
- From a turning point to a turning point $(x_{1}, x_{2}), (x_{2}, x_{3})$
- From a turning point to an end behavior $(x_{3}, + \infty)$
Then add a description of the x-values for each interval as increasing, decreasing, or constant.

Interval Description	x	y	Interval Notation
x- and y-values increase across the interval	↑	↑	increasing: $(n_{1}, + \infty)$
x-values increase and y-values decrease across the interval	↑	↓	decreasing: $(- \infty, n_{2})$
x-values increase and y-values remain the same	↑	↔	constant: $(n_{1}, n_{2})$

Note

These descriptions should sound familiar. In linear functions, they describe the rate of change (or slope) as well. While these graphs are not linear, they can be described generally as increasing, decreasing, or constant.

Depending on the problem, you may also be asked to identify other characteristics from the graph.

How to Express Interval Descriptions

The graph of the polynomial function decreases for x-values across the intervals $(- \infty, x_{1})$ and $(x_{2}, x_{3})$ . 
The graph of the polynomial function increases for x-values across the intervals $(x_{1}, x_{2})$ and $(x_{3}, + \infty)$ .

Example 5

Name the interval(s) in which the function increases and decreases.

The graph of the polynomial function increases across the interval $(- 2.431, 1.097)$ .
The graph of the polynomial function decreases across the intervals $(- \infty, - 2.431)$ and $(1.097, + \infty)$ .

Note

If you want to explore this function more on your own using technology, the equation for the graph is: $f (x) = - x^{3} - 2 x^{2} + 8 x + 2$

Example 6

$h (x) = x (x + 1) (x - 3) (x - \sqrt{5}) (x + \sqrt{5})$

Name the interval(s) in which the function increases and decreases.
Name the relative minimum and maximum over the interval $[- 3, 1]$ .

x	y
–1.796	12.166	↑
–0.509	–4.158	↓
1.226	16.93	↑
2.678	–6.888	↓

Increases across the intervals: $(- \infty, - 1.796), (- 0.509, 1.226), (2.678, + \infty)$
Decreases across the intervals: $(- 1.796, - 0.509), (1.226, 2.678)$
Relative minimum: $(- 0.509, - 4.158)$
Relative maximum: $(- 1.796, 12.166)$

Note

Notice the relative minimum and maximum points are not the highest and lowest points on the graph, but they are the relative minimum and maximum for the given interval, $[- 3, 1]$ .