Main content

### Course: Integrated math 3 > Unit 6

Lesson 2: Reflecting functions# Reflecting functions: examples

We can reflect the graph of any function f about the x-axis by graphing y=-f(x) and we can reflect it about the y-axis by graphing y=f(-x). We can even reflect it about both axes by graphing y=-f(-x). See how this is applied to solve various problems.

## Want to join the conversation?

- I don’t exactly understand how f(-x) and -f(x) get different results.(10 votes)
- f(-x) multiplies all the x in the equation by negative 1 whereas negative f would make the entire equation negative.(8 votes)

- This is super hard.Find g(x), where g(x) is the reflection across the x-axis of f(x)=–4(x+7)2–7. Which of these do i change?(5 votes)
- We know these two things:

-f(x) = Reflection across x-axis

f(-x) = Reflection across y-axis

These are essential to know.

Since -f(x) is a reflection across axis, we do as such:

-f(x) = -1(-4(x+7)²-7)

-f(x) = 4(x+7)²+7 <-- you're new equation(9 votes)

- Hello, sorry if someone asked this already, but why is √x a curved line? Thanks.(5 votes)
- the simple answer is it is curved because it is not linear (it is either linear or curved). So linear functions and absolute value functions (and step functions) are not curved, but many other functions are curved including square root.

Since perfect square roots get further apart every time, the graph goes up slower and slower between perfect squares which creates this curve. (1,1)(4,2)(9,3)(16,4) slope between first two points is 1/3, slope between second and third is 1/5, slope between third and fourth is 1/7, etc. Linear equations would have same slope between any two points on the line.(7 votes)

- what's the difference when the exponent is in parenthesis and when it's not? like difference between 1/8*4^(2-x) vs 1/8*4^2-x(4 votes)
- It all comes down to order of operations.

Exponentiation takes precedence over subtraction,

but parentheses take precedence over exponentiation.

So, 4^3 − 1 tells us to do the exponentiation before the subtraction,

while 4^(3 − 1) tells us to do the parenthesis first, which means we do the subtraction before the exponentiation.

4^3 − 1 = 64 − 1 = 63

4^(3 − 1) = 4^2 = 16(4 votes)

- If the function y=x−−√ is replaced by x=y√, then the new graph can be described as a reflection of y=x(4 votes)
- What you have is confusing and not a function or an equation, you have minus a negative square root with nothing in the root, then you change it by leaving off the minus negative, but still have a root symbol without anything inside. If you do not have a function in the first place, there is no reflecting across anything.(4 votes)

- 1:16

What does Sal mean by g(0)?(2 votes)- g(0) simply means "the function g(x) evaluated at x = 0"(6 votes)

- Why does f(x) = the square root of x make the type of graph it does?(2 votes)
- When you take the inverse of a function, its graph gets reflected across the line y=x. The graph of y=√x is half of a parabola, opening sideways, since it's a reflection of y=x².(3 votes)

- How do you rotate a function?(3 votes)
- i still mix between the reflection over y-axis and reflection over x-axis. could someone help please(0 votes)
- Try making a visual or mental picture of these reflections. From this picture, you can make the following observations:

1) two points that are mirror images of each other about the y-axis have the same y-coordinate and x-coordinates that are opposites of each other, and

2) two points that are mirror images of each other about the x-axis have the same x-coordinate and y-coordinates that are opposites of each other.(7 votes)

- Sal tells us that we want to graph a new function, g(x), that transforms an old one, f(-x).

I interpret this as meaning that the Y value [g(x)] changes… because the term g(x) [or f(x)] is often used as a synonym for the Y value (i.e. the output) of an equation.

Thus, I thought that if the Y value of f(x) was one, then the Y value of g(x) will be -1. This would flip the graph around the X axis.

But instead Sal shows the new g(x) figure as being flipped around the Y axis.

Why am I getting this backward?

Is it because the value within parentheses in f(x) refers not to the output Y value but actually/only to the input X value?(2 votes)- Not quite, the only way this would be true is if f(x)=x. Let's say f(x)= 3x + 4, then g(x) = f(-x) = -3x+4. Thus, if f(x)=1, that would mean 1=3x+4, x=-1. g(-1)=-3(-1)+4=7.

A negative outside reflects across x, so that would be -f(x). A negative inside reflects across y, which is what is shown f(-x).

Hope this helps.

I am not sure how to answer your very last question, if f(x)=1, that refers to some value of x will give an output of 1. So outside would maintain the x value of the point with opposite signs, and inside would maintain the y value of the function (signs could be anything), not the x value.(3 votes)

## Video transcript

- [Instructor] What we're
going to do in this video is do some practice examples
of exercises on Khan Academy that deal with reflections of functions. So this first one says this
is the graph of function f. Fair enough. Function g is defined as g of
x is equal to f of negative x. Also fair enough. What is the graph of g? And on Khan Academy, it's multiple choice, but I thought for the sake of this video, it'd be fun to think about
what g would look like without having any choices,
just sketching it out. So pause this video and
try to think about it, at least in your head. All right, now let's work
through this together. So we've already gone
over that g of x is equal to f of negative x. So whatever the value of
f is at a certain value, we would expect g to take on that value at the negative of that. So for example, we can see
that f of four is equal to two, so we would expect g of
negative four to be equal to two because, once again, g of negative four, we could write it over here. G of negative four is going to be equal to f of the negative of negative four, which is equal to f of four. And so we could keep going with that. What would g of negative two be? Well, that would be the
same thing as f of two, which is zero, so it
would be right over there. What would g of zero be? Well, that would be the
same thing as f of zero 'cause a negative zero is zero. And f of zero is right over there. Looks like negative two. And so you can already
see where this is going. And we've already talked
about it in previous videos that if you replace your
x with a negative x, you're essentially
reflecting over the y-axis. So g is going to look something like this. It is going to look something like this. Once again, g of negative
six would be the same thing as f of six. And so that would be the graph of g. And if you're doing this on Khan Academy, you'd pick the choice
that looks like this, that would give a
reflection over the y-axis. Let's do another example. So here once again, this is
the graph of the function f. And then they say what is the graph of g? And so pause this video and at least try to sketch it out in your
mind what g should look like. All right, so in this situation, they didn't replace the x
with negative x in f of x. Instead, g of x is equal to
the negative of all of f of x. In fact, we could
rewrite g of x like this. We could say that g of
x is equal to, notice, all of this right over here, that was our definition of f of x. So g of x is equal to
the negative of f of x. So instead of it being f of negative x, it's equal to the negative of f of x. So one way to think about it is we can see that f of zero is two, but g of zero is going to
be the negative of that. So it's going to be equal to negative two. And so you could keep going with that. You could see that whatever
f of a certain value is, g of that value would
be the negative of that. So it would be down here. And so g of x would be
a reflection of f of x about the x-axis. So g of x is going to look something like that, a reflection about the x-axis. Once again, on Khan Academy,
you'd pick the choice that would actually look like that. Let's do another example. His is strangely fun. (laughs) All right, so here
we're told functions f, so that's in solid in this blue color, and g dashed, so that's right
over there, are graphed. What is the equation of g in terms of f? So pause this video and
try to think about it. So the key is to realize
how do we transform f of x, actually, they've labeled it over here, this is f of x right over
here, in order to get g? So f of negative x would be a reflection of f about the y-axis. And so it would intersect there. It would have this
straight portion like this. And I'm just experimenting right now. It'd have the straight portion like this. And then it would go up. And let's see. If f of negative x. So when you input six into it, that would be f of
negative six, which is six. So it would go up there. So f of negative x would
look something like this. Something like that. So the purple is f of negative x. Now, that doesn't quite get us to g, but it gets us a little bit closer 'cause it looks like if I
were to take the reflection of f of negative x, f of
negative x about the x-axis, it looks like I'm going to get to g. And so how do you reflect
something about the x-axis? Well, we saw it in the example just now. You multiply the entire
function by a negative. So we could say that g is equal to the negative of f of negative x. It's equal to the negative of this. So we're doing both reflections. We're flipping over the y-axis, and we're flipping over
the x-axis to get to g. Let's do one more example. So once again, they've
graphed f, they've graphed g, and they've said f is defined
as this right over here. What is the equation of g? So they're not just
asking it in terms of f. They just wanna know what
is the equation of g? Pause this video and
try to think about it. Well, you can see pretty clearly that this is a reflection
across the y-axis. And a reflection across the y-axis, you can see pretty clearly, that g of x is equal to f of negative x. F of negative x. How do we know that? Well, for whenever we take f
of x and we get that value, g at the negative of that value takes on the same function
value, I guess I could say. Or another way to think about it is we could just pick this
point, negative eight. F of negative eight is
equal to a little over four, but g of eight is equal
to a little over four, is equal to that same value. And so what is the equation of g? Well, we just have to rewrite this so that we can write
it out as an equation. And so we could write
out g of x is equal to. If I were to replace all of the x's here with a negative x, what would I get? I would get four times the
square root of two minus. Instead of an x, I will have a negative x. And then the minus eight
is outside of the radical. And so we would have g of x is equal to four times the square root
of two plus x minus eight. And we're done.