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## Algebra 2

# Symmetry of algebraic models

CCSS.Math: ,

Sal interprets the significance of modeling function being even. Created by Sal Khan.

## Want to join the conversation?

- in the following questions does anyone know what the term end behavior means?(5 votes)
- Yes, end behavior is how a function behaves as x gets very large (both in the negative and positive direction). The main thing we are interested in is whether f(x) is increasing or decreasing as x grows very large.

For example, f(x) = x² - x + 2 keeps increasing as x gets very large, so its positive end behavior is "increasing" or "upward" (different teachers use different word".

Likewise, when x becomes very negative, this function keeps increasing so its negative end behavior is "increasing" or "upward".

But, f(x) = x³ + 2x² - 3 is different. When x is very negative, it is decreasing or "downward". So its negative (sometimes called left hand) end behavior is downward or decreasing. However, when x if very large on the positive side, f(x) goes up, so its positive or right hand end behavior is increasing or upward.

If you have a good algebra teacher, you will get a slightly more rigorous way of saying the same thing:

Negative side, going down would be written as: f(x)→−∞ as x→−∞

This is read as "f(x) approaches negative infinity as x approaches negative infinity"

Similarly,

Negative side, going up would be written as: f(x)→ + ∞ as x→−∞

Positive side, going down would be written as: f(x)→−∞ as x→ +∞

Positive side, going up would be written as: f(x)→ + ∞ as x→+∞

There is a video on this:

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/polynomial-end-behavior/v/polynomial-end-behavior(5 votes)

- At1:08, Sal says that T(v) = T(-v) defines the function as even. An odd function is defined as T(v) = -T(-v), right?(3 votes)
- This is not super necessary to say but I just noticed that... Scott, you're right T(v)= -T(-v) but Gene is also right -T(v)=T(-v). Those two statements say the same thing. To convert from one to the other just multiply both sides by -1.(5 votes)

- So magnitude is like absolute value?(3 votes)
- Well, absolute value is a type of magnitude (amount given, but no direction given).

On the other hand, any given integer (signed number) represents a vector (amount and a direction is given). Not sure if that answers your question...(2 votes)

- Did khan academy change their videos to their own? I liked the youtubes(2 votes)
- So any quadratic function who's symmetry is the y axis is even?

EDIT: And any absolute value function who's symmetry is the y axis is even?(1 vote)- Any function that is symmetric about the y axis is even, period. It doesn't matter if it's polynomial, trigonometric, or whatever weird thing you can cook up.(2 votes)

- At1:08, How did they figure out the equation to use so you can get the answer? What would you use to get the equation?(1 vote)
- Could somebody explain in more detail?(1 vote)
- Wouldn't the function also be dependent of time? If you just sand for 3 seconds, it won't do much, but if you sand for an hour, it is a lot of sanded wood.(1 vote)

## Video transcript

Cid is experimenting
with a piece of sandpaper and some wood. He tries scraping the
piece of sandpaper over the wood in different ways
to see how much is scraped off. The thickness of wood
scraped off, in millimeters, as a function of the
speed of the sandpaper, in meters per second. t of v. So this is the
thickness scraped off. So that's the thickness,
or how much is scraped off. And it is a function of speed. One, they're using v and also
they're getting negative value. So we care about the direction. It's actually the velocity. So this is how much is scraped
off as a function of velocity. It's shown below. And so if the velocity
is greater than 0, that means that the sandpaper
is moving to the right. That makes sense. That's the standard convention. And if the velocity
is less than 0, it means the sandpaper
is moving to the left. Fair enough. The function is even. What is the significance of
the evenness of this function? Well, the fact that it's
even means that t of v is equal to t of negative
v. So that tells us that if our velocity is 8
meters per second to the left we're going to get
as much scraped off as if we go 8 millimeters
per second to the right. And we see that right over here. So that is equal to that. If we go at 6 meters
per second to the left we're going to get just
as much scraped off as we go 6 millimeters--
6 meters per second, these are in meters per
second-- to the right. So these two are
going to be the same. So it's really telling
us-- and we could say do it for 4 meters per
second and negative 4-- is it doesn't matter if we
go to the left or the right. What really matters is the
magnitude of the velocity or the absolute value of it. But it doesn't matter if we're
going to the left or the right. Whether we're going to
the left or the right for a given
magnitude of velocity we are going to get the
same amount scraped off. Now let's see which
of these choices are consistent with
what I just said. Moving the sandpaper faster
scrapes off more wood. Well, that's true. We see as the speed increases,
or the magnitude of the speed increases, we scrape
off more wood. As the magnitude of the
speed, this negative 8, you might say, hey, that's
lower than negative 2, but the magnitude is larger. We're going 8 meters
per second to the left and we're scraping off more. So this is a true
statement, but it's not the significance of the
evenness of the function. This could have been true
even if this was a seven, but then this function
would no longer be even. The piece of wood is
six millimeters thick. We actually don't get any
of that from the function. Moving the sandpaper
to the right has the same effect as
moving it to the left. Well, that seems pretty close
to what I had said earlier. That for a given speed to
the right or to the left, we get the same amount
that is taken off of the piece of sandpaper,
or the piece of wood. So this looks like our answer. Keeping the sandpaper still
doesn't scrape off any wood. Well, that is true
but once again is not the significance of
the evenness of this function.