Understanding Positive/Negative Intervals & Increasing/Decreasing Intervals

• NeomiXD
In summary, Positive and negative intervals refer to the sections on a graph where the function has a positive or negative slope, respectively. They are determined by looking at the slope of the function at different points and can greatly affect the overall behavior of the function. A function can have both positive and negative intervals, which can help determine critical points and the overall trend of the function.

NeomiXD

I do not understand positive and negative intervals and increasing/decreasing intervals. I included 2 examples from my textbook which I did not understand and I was wondering if someone can explain it to me.

Example 1:

f(x) = 2 - x

x intercept is (2, 0) and y-intercept is (0,2)

f(x) is positive when xε(-∞, 2) and negative xε(2, ∞)
f(x) is decreasing when xε(-∞, ∞)

reciprocal od the function is g(x) = 1 / 2 - x

reciprocal function is positive when xε(-∞, 2) and negative xε(2, ∞)
It is increasing when xε(-∞, 2) and when xε(2, ∞)

Example 2:

f(x) = 9 - x^2

x-intercepts are 3 and -3

f(x) is positive when xε(-3, 3) and negative when xε(-∞, -3) and when xε(3, ∞)
f (x) is increasing when xε(-∞, 0) and decreasing when xε(0, ∞)

g(x) = 1 / 9- x^2

reciprocal function is positive when xε(-3, 3) and negative xε(-∞, -3) and when xε(3, ∞)
It is decreasing when xε(-∞, -3) and when xε( -3, 0) and increasing when xε(0, 3) and when xε(3, ∞)

NeomiXD said:
Example 1:

f(x) = 2 - x

x intercept is (2, 0) and y-intercept is (0,2)

f(x) is positive when xε(-∞, 2) and negative xε(2, ∞)
xε(-∞, 2) means x < 2. When x < 2, f(x) > 0.
xε(2, ∞) means x > 2. When x >2, f(x) < 0.
f(x) is decreasing when xε(-∞, ∞)
When you look at the graph from left to right, the x values are increasing. As the x values increase, the corresponding f(x) values are decreasing.
NeomiXD said:
Example 2:

f(x) = 9 - x^2

x-intercepts are 3 and -3

f(x) is positive when xε(-3, 3) and negative when xε(-∞, -3) and when xε(3, ∞)
f (x) is increasing when xε(-∞, 0) and decreasing when xε(0, ∞)
xε(-∞, 0): for all values "between" -∞ and 0 (ie. the left side of the graph), as you look left to right, the corresponding f(x) values are increasing.
xε(0, ∞): for all values "between" 0 and ∞ (ie. the right side of the graph), as you look left to right, the corresponding f(x) values are decreasing.

1. What are positive and negative intervals?

Positive and negative intervals refer to the sections on a graph where the function has a positive or negative slope, respectively. In other words, positive intervals are where the function is increasing, while negative intervals are where the function is decreasing.

2. How do you determine positive and negative intervals from a graph?

To determine positive and negative intervals from a graph, you need to look at the slope of the function at different points. If the slope is positive, the interval is positive, and if the slope is negative, the interval is negative. You can also look for the points where the graph crosses the x-axis - if the graph crosses from below to above, it is a positive interval, and if it crosses from above to below, it is a negative interval.

3. How are positive and negative intervals related to increasing and decreasing intervals?

Positive and negative intervals are directly related to increasing and decreasing intervals. A positive interval is where the function is increasing, and a negative interval is where the function is decreasing. This means that the function is getting larger in value during a positive interval and getting smaller during a negative interval.

4. Can a function have both positive and negative intervals?

Yes, a function can have both positive and negative intervals. This is because a function can have sections where it is increasing and sections where it is decreasing, depending on the slope of the function at different points.

5. How do positive and negative intervals affect the overall behavior of a function?

Positive and negative intervals can greatly affect the overall behavior of a function. They can tell you whether the function is increasing or decreasing, which can give you an idea of the overall trend of the function. They can also help you determine the critical points of a function, such as the maximum and minimum values. Understanding positive and negative intervals is crucial for analyzing and graphing functions.