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## 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

CCSS.Math:

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?(26 votes)
- 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.(44 votes)

- does 0 count as positive or negative?(6 votes)
- 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!(45 votes)

- At2:16the sign is little bit confusing. More explanation. Thanks(6 votes)
- Sal wrote b < x < c. Between the points b and c on the x-axis, but
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.**not**(12 votes)

- Wouldn't point a - the y line be negative because in the x term it is negative?(8 votes)
- 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.(5 votes)

- 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)?(7 votes)
- @celestec1, I do not think there is a y-intercept because the line is a function. This is just based on my opinion(2 votes)

- So zero is not a positive number?(2 votes)
- Correct. Zero is the dividing point between positive and negative numbers but it is neither positive or negative.(9 votes)

- This linear function is discrete, correct?(2 votes)
- No, this function is neither linear nor discrete. It is continuous and, if I had to guess, I'd say cubic instead of linear.(6 votes)

- 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.(3 votes)- 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.(2 votes)

- 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.(2 votes)- If you have a x^2 term, you need to realize it is a quadratic function. There is no meaning to increasing and decreasing because it is a parabola (sort of a U shape) unless you are talking about one side or the other of the vertex. Increasing and decreasing sort of implies a linear equation.

If it is linear, try several points such as 1 or 2 to get a trend.(4 votes)

- What does the variable f stand for?(2 votes)
- If you are referring to the use of "f" in the video, it tells you the graph represents y = f(x).

This means the variable "y" equals the function called "f" which is defined with the values of "x" as the inputs to the function. The actual function is depicted in the graph.(3 votes)

## Video transcript

- [Voiceover] What I
hope to do in this video is look at this graph y is equal to f of x and think about the intervals where this graph is positive or negative and then think about the intervals when this graph is
increasing or decreasing. So first let's just think
about when is this function, when is this function positive? Well positive means that the value of the function is greater than zero. It means that the value
of the function this means that the function is
sitting above the x-axis. So it's sitting above the x-axis in this place right over here that I am highlighting in
yellow and it is also sitting above the x-axis over here. And if we wanted to, if we wanted to write those intervals mathematically. Well let's see, let's say that this point, let's say that this point
right over here is x equals a. Let's say that this right
over here is x equals b and this right over here is x equals c. Then it's positive, it's
positive as long as x is between a and b. At x equals a or at x equals b the value of our function is zero but it's positive when x is between a and b, a and b or if x is greater than c. X is, we could write it
there, c is less than x or we could write that
x is greater than c. These are the intervals when
our function is positive. Let me write this, f of x, f of x positive when x is in this
interval or this interval or that interval. So when is f of x negative? Let me do this in another color. F of x is going to be negative. Well, it's gonna be negative
if x is less than a. So this is if x is less than
a or if x is between b and c then we see that f of
x is below the x-axis. F of x is down here so this
is where it's negative. So here or, or x is between
b or c, x is between b and c. And I'm not saying less than or equal to because at b or c the value of
the function f of b is zero, f of c is zero. That's where we are actually
intersecting the x-axis. So that was reasonably straightforward. Now let's ask ourselves
a different question. When is the function
increasing or decreasing? So when is f of x, f of x increasing? Well increasing, one way to
think about it is every time that x is increasing then
y should be increasing or another way to think
about it, you have a, you have a positive rate of
change of y with respect to x. We could even think about it as imagine if you had a tangent line
at any of these points. If you had a tangent line at
any of these points the slope of that tangent line is
going to be positive. But the easiest way for
me to think about it is as you increase x you're
going to be increasing y. So where is the function increasing? Well I'm doing it in blue. So it's increasing right until we get to this point right over
here, right until we get to that point over there
then it starts decreasing until we get to this
point right over here and then it starts increasing again. It starts, it starts increasing again. So let me make some more labels here. So let's say that this,
this is x equals d and that this right over here, actually let me do that in green color, so let's say this is x equals d. Now it's not a, d, b but you
get the picture and let's say that this is x is equal to, x is equal to, let me redo it a little
bit, x is equal to e. X is equal to e. So when is this function increasing? Well it's increasing if x is
less than d, x is less than d and I'm not gonna say
less than or equal to 'cause right at x equals
d it looks like just for that moment the
slope of the tangent line looks like it would be,
it would be constant. We're going from increasing to decreasing so right at d we're neither
increasing or decreasing. But then we're also increasing,
so if x is less than d or x is greater than e,
or x is greater than e. And where is f of x decreasing? So f of x, let me do this
in a different color. When is, let me pick a
mauve, so f of x decreasing, decreasing well it's going
to be right over here. It's gonna be right between d and e. Between x equals d and x equals e but not exactly at those points 'cause at both of those points you're neither increasing nor decreasing but you see right over
here as x increases, as you increase your x
what's happening to your y? If you go from this point
and you increase your x what happened to your y? Your y has decreased. You increase your x, your y has decreased, you increase your x, y has decreased, increase x, y has decreased all the way until this point over here. So f of x is decreasing
for x between d and e. So hopefully that gives
you a sense of things. Notice, these aren't the same intervals. That we are, the intervals
where we're positive or negative don't perfectly coincide with when we are increasing or decreasing. So it's very important to
think about these separately even though they kinda sound the same.