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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus>Unit 5

Lesson 2: Intervals where a function is positive, negative, increasing, or decreasing

# Increasing, decreasing, positive or negative intervals

Function values can be positive or negative, and they can increase or decrease as the input increases. Here we introduce these basic properties of functions.

## Want to join the conversation?

• I have a question, what if the parabola is above the x intercept, and doesn't touch it? Is there not a negative interval?
• That is true, if the parabola is upward-facing and the vertex is above the x-axis, there would not be an interval where the function is negative. You have to be careful about the wording of the question though. The function would be positive, but the function would be decreasing until it hits its vertex or minimum point if the parabola is upward facing. If the function is decreasing, it has a negative rate of growth. In other words, while the function is decreasing, its slope would be negative. You could name an interval where the function is positive and the slope is negative. The secret is paying attention to the exact words in the question.
• does 0 count as positive or negative?
• That's a good question! Some people might think 0 is negative because it is less than 1, and some other people might think it's positive because it is more than -1. But in actuality, positive and negative numbers are defined the way they are BECAUSE of zero. If a number is less than zero, it will be a negative number, and if a number is larger than zero, it will be a positive number. So zero is actually neither positive or negative.
Zero can, however, be described as parts of both positive and negative numbers. For example, if someone were to ask you what all the non-negative numbers were, you'd start with zero, and keep going from 1 to infinity. (0, 1, 2, 3, 4...to infinity) Alternatively, if someone asked you what all the non-positive numbers were, you'd start at zero and keep going from -1 to negative-infinity. (0, -1, -2, -3, -4 ... to -infinity)
BUT what if someone were to ask you what all the non-negative and non-positive numbers were? Well, then the only number that falls into that category is zero!
• At the sign is little bit confusing. More explanation. Thanks
• Sal wrote b < x < c. Between the points b and c on the x-axis, but not including those points, the function is negative. Notice, as Sal mentions, that this portion of the graph is below the x-axis. That is your first clue that the function is negative at that spot. Hope this helps.
• Wouldn't point a - the y line be negative because in the x term it is negative?
• No, the question is whether the function f(x) is positive or negative for this part of the video. That means, according to the vertical axis, or "y" axis, is the value of f(a) positive --is f(x) positive at the point a? In other words, what counts is whether y itself is positive or negative (or zero).

At point a, the function f(x) is equal to zero, which is neither positive nor negative. It makes no difference whether the x value is positive or negative.
• So zero is not a positive number?
• Correct. Zero is the dividing point between positive and negative numbers but it is neither positive or negative.
• If a function is increasing on the whole real line then is it an acceptable answer to say that the function is increasing on (-infinity, 0) and (0, infinity)?
• @celestec1, I do not think there is a y-intercept because the line is a function. This is just based on my opinion
• So...How do we know if the interval is increasing or decreasing?? I STILL don't get it...Can anyone explain this to me in a simpler and shorter way?
• Think in terms of slopes like with linear equations. When a line has a negative slope, it moves downward as the line moves left to right. If a line has a positive slope, then it moves upwards as the line move left to right.

Now, apply these same ideas to other types of graphs. If the graph is moving downward, then that is a decreasing interval. If the graph is moving upward, then it is a increasing interval.

Hope this helps.
• This linear function is discrete, correct?
• No, this function is neither linear nor discrete. It is continuous and, if I had to guess, I'd say cubic instead of linear.

• Why OR? Shouldn’t it be AND?
• OR means one of the 2 conditions must apply
AND means both conditions must apply for any value of "x"

For example, in the 1st example in the video, a value of "x" can't both be in the range a<x<b and also in the range x>c. This is why OR is being used.

Hope this helps.
• f(x)= x^2-4x
I multiplied 0 in the x's and it resulted to f(x)=0? Is this right and is it increasing or decreasing... I'm slow in math so don't laugh at my question.
• I'm not sure what you mean by "you multiplied 0 in the x's". If you mean that you let x=0, then f(0) = 0^2-4*0 then this does equal 0. So, f(0)=0. This function decreases over an interval and increases over different intervals.