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Problem 48 Hard Difficulty

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

(a) $ y = \ln (-x) $
(b) $ y = \ln \mid x \mid $

Answer

a) See explanation, see graph
b See explanation

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Video Transcript

to make a rough sketch of this function. Let's start by thinking about the standard function. Why equals natural log of X? So we have an idea of what that looks like. We have log rhythmic growth. We have an X intercept at 10 We know it goes through the point e one. So then what effect does multiplying the X by a negative have on the graph that's going to reflect it across the Y axis. So let's draw what that would look like. So the 0.10 is going to reflect across to the point negative 10 and the point e one is going to reflect across to the point negative E one. So our graph looks like this. And for part B again, we can start with y equals natural log of X just to get the basics down, going through the 0.10 going through the point e one. Now, how about the absolute value? What effect is that going to have on it? Well, previously, we couldn't substitute any negative X values into the natural log function. You could only take the natural log of a positive. However, if we're taking the absolute value before we find the Y coordinate. Now we can use negatives. So if X was negative one, we would be finding the natural log of positive one and effects was negative. Two. We would be finding the natural log of positive two. We would take the absolute value. That should say positive. Two. We would take the absolute value before we put it into the natural log function. So the value we get for negative one is going to be the same as the value we get for one. The value we get for negative two is going to be the same as the value we get for two etcetera. So we're going to have points on the left side of the Y axis that reflect the points that were on the right side of the Y axis. So we will still have the graph. It was on the right side. We will also have a piece on the left side and they should be mirror images of each other