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### Course: Algebra 2>Unit 9

Lesson 2: Reflecting functions

# Reflecting functions introduction

We can reflect the graph of y=f(x) over the x-axis by graphing y=-f(x) and over the y-axis by graphing y=f(-x). See this in action and understand why it happens.

## Want to join the conversation?

• Khan wants to accentuate some of those curves
• how did Desmos take the sqr(-x)? an imaginary number in a two dimensional plane doesn't make sense to me.
• It is not imaginary for the whole domain. If you put a 0 in, it is real. Further, if you put in negative values for x, - (-x) gives a positive x. So adding this negative creates a relection across the y axis, and the domain is x ≤ 0.
• I thought it was not possible to graph sqrt(-1) unless I use imaginary numbers, is this graphing website consistent?
• Yeah, it is. You can tell because when you graph sqrt(x) the first quadrant is empty because plotting sqrt of negative numbers isn't possible without imaginary numbers. If you plot sqrt(-x), the second quadrant is instead, because the first quadrant is now sqrt of positive numbers (negative * negative = positive.) :)
• How can I tell whether it's flipping over the x-axis or the y-axis (visually speaking)
• Like other functions, f(x) = a g(bx), if a is negative (outside) it reflects across x axis and if b is negative it reflects across the y axis. So for square root functions, it would look like y = a √(bx). Outside reflect across x such as y = -√x, and inside reflect across y such as y = √-x. It works for all functions though many reflections will not look different based on the function. Quadratic y = -x^2 reflects across x, y = (-x)^2 reflects across y (though it would be the same because of reflexive property of quadratics).
• How is it possible to graph a number which seemingly never ends (like e at )?
• As far as I know, most calculators and graphing applications just have a built-in set approximation for common irrational numbers like e, calculated beforehand from a definition like the infinite sum of (1/n!). We can't really know what e is, besides e itself, so we use an approximation instead of calculating e to a billion places for every point we use in the graph, to save computing power.
• for the k(x) shouldnt the 2 negatives cancel each other out and become a positive?
• That is when they're multiplied directly against each other. That does not apply when, let's say, an nth (i.e a square) root or an absolute value is in between it, like for k(x).
• What is a principle root?
• Most numbers have two roots, a number and its negative inverse. For example, the square roots of 25 are 5 and -5, since (5*5) = (-5*-5) = 25.
The principle root of a number is simply the positive root, which in the previous example is 5.
Hope this helps