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### Course: Algebra 2>Unit 9

Lesson 8: Graphs of logarithmic functions

# Graphing logarithmic functions (example 1)

We can graph y=2log₂(-x-3) by viewing it as a transformation of y=log₂(x).

## Want to join the conversation?

• What is the use of changing y=2log₂(-x-3) to y=2log₂(-(x+3)) ? Even shifting graph by 3 to the left doesn't really make a sense to me, cuz y=2log₂(-(x+3)) should end up in the previous form y=2log₂(-x-3) and then we should have shifted by 3 to the right instead ... I don't know but I am stuck :(
• It's easier to understand by first looking at a simpler transformation. If you have y=x^2, and you shift to y=(x-3)^2, you've shifted the graph three to the right. Why? For y=x^2, when x=0, y=0. For y=(x-3)^2, where does y=0? At x=3. I.e., in the new graph, the old vertex (which was at x=0) is now at x=3, hence the graph has shifted 3 to the right. Similar principle in this video's equations -- we now have to change x to '-3' to get the same result as when x was '0' before, so the new function has moved over 3 to the left. It's easiest to see that when it's written -(x+3) rather than (-x-3). -(x+3) is also better because it more clearly displays the horizontal transformation that's taking place by pulling out that -, since -(x...) represents a horizontal transformation.
• Should a negative always be factored out when both the x and a constant are negative such as in: (-x-3) to -(x+3) to help with a transformation?
(-x+3) to -(x-3) or
(x-3) to -(-x+3) or
(x+3) to -(-x-3)
• I played about with demos graphing calculator and basically the answer to my own question is: we dont want the x to be negative if we want to assume the correct shifting behavior. Meaning, (x+3) is a shift to the left 3 or (x-3) is a shift to the right three but having the negative on the x such as (-x+3) does the opposite of what we should expect (this shifts the graph to the right 3), Thus (-x+3) or (-x-3) the negative should be factored out to assume regular behavior.
• At around , Sal says that the function moves 3 to the left. Why? It doesn't really make sense to me because it is in the form of -(x+3). I get that you need to shift 3 to the left but isn't there a negative sign at the front? Doesn't that have any influence on the shifting? I'm very confused.
• The -(x+3) is done so that it equals (-x-3) in the equation. Trying to graph (-x-3) will not work, because the negative sign in front of the x stops us from making any transformations. Think about it as g(x) = f(-(x+3)). If we just look at the negative part, as in g(x) = f(-x), the graph will get flipped over the x axis. If we look at the other half, so g(x) = f(x+3), and we take x as 5, then g(5) = g(8). In this way, if you map it out, the entire graph is shifted left. Both of these transformations result in f(-(x+3)). Let me know if you have further questions!
• I got lost about when Sal started talking about -(x+3) x--3 and he says that moves the graph to the left. I thought negative numbers moved the graph to the right?
• Why does Sal say at that the y values in y = log_2(-x) are the same as when the x values were positive? He then says that the "log base two of the negative of negative four, well that's still log base two of four, so that's still going to be two", and he doesn't say where the second negative came from.
• At the second negative came from the fact that 2 negatives equal a positive.
In other words: --=+.
Hope this helps.
• Does it matter what order I pick apart my function in? Like, if I did the times 2 first and then the -x and then shifted 3 units to the left?
• I have to say I am pretty sure that the order does not matter because each transformation does something unique (that is they are independent of each other). Note that 3 units to left is based on factoring out the negative inside the log.
• If I have an equation like: -2log₂(x+3), Should I reflect the graph over the x axis first (because of the negative sign), or multiply my y values first, or multiple my y values by negative 2? Can someone please explain this transformation to me, I'm very confused.

Thank You!
• All three options will give the same end result because of the commutative property of multiplication. So, reflecting the graph and multiplying by 2, multiplying by 2 and reflecting the graph, or multiplying by -2 are all valid.
• he forget the apply reflective transformation which is indeed when there is a "minus" in front of (x+3) variable.