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### Course: Algebra 2>Unit 9

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.
• f(-x) multiplies all the x in the equation by negative 1 whereas negative f would make the entire equation negative.
• 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?
• 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
• Hello, sorry if someone asked this already, but why is √x a curved line? Thanks.
• 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.

• What does Sal mean by g(0)?
• g(0) simply means "the function g(x) evaluated at x = 0"
• 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
• 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
• If the function y=x−−√ is replaced by x=y√, then the new graph can be described as a reflection of y=x
• 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.
• How do you rotate a function?
• Why does f(x) = the square root of x make the type of graph it does?
(1 vote)
• 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.
• so for g(x)= f(-x) we keep the same y value but make the x val negative
and for g(x)= -f(x) we keep the same x value but make the y value negative

Then is the f in f(x) supposed to represent the y value?
(1 vote)
• No, not necessarily. f(x) stands for the function of x or y in terms of x. Which you can think of as a symbol or name that tells us what to do with x. Inside the parentheses of f(x), is the value of x, and f(x) is the value of y.
So, when you have -f(x), you're not turning just the f into negative, but the entire -(f(x)), and the "f" is there of describe the specific function.

TDLR: The f(x) respresents the y value, not just the f.