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Question 202833: I am sapose to graph y=1/(x-3) and y= -2|x|
They are both really confusing to me and i dont know how to go about it.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I'll start with the "easy" one: y = -2|x|. You probably know how to figure out absolute values. The absolute value of 0 is 0 (which is the same as the number). The absolute value of a positive number is the same as the number. The absolute value of a negative number is the negative of that negative number (the negative of a negative is a positive).
Now let's write this logic in Mathematical notation:
|x| = x if x>=0
|x| = -x if x < 0
Think about this. Once we understand this we can think of our equation with absolute value as two equations without absolute value:
y = -2x if x>=0 and y = -2*(-x) = 2x if x < 0.
So we can graph y = -2|x| by graphing y = -2x for x>=0 (which is from the y-axis and going to the right) and graphing y = 2x for x < 0 (which is left of the y-axis). y = -2x and y = 2x are simple lines and should be easy to graph. The "trick" is to graph y = -2x only on the y-axis (x = 0) and to the right of the y-axis (x > 0). And then graph the y = 2x line only to the left of the y-axis). The graph should look like the graph below:
The other equation, y = 1/(x-3), is a bit more difficult. I hope you know what asymptotes are. In short, asymptotes are lines which the graph of a function gets closer and closer to but rarely ever cross. Graphs of functions represent the points which fit the equation. Asymptotes are not part of the graph of the function because points on these lines do not fit the equation. Because of this we draw the asymptotes as dotted lines indicating that the lines are not part of the graph. We use these dotted lines to help us sketch the graph of the actual function (as I hope you will understand shortly).
Vertical lines, as I hope you understand) represent all points for a certain x-value. x=5 is a vertical line which passes through 5 on the x-axis. Vertical asymptotes occur for x-values which make a function's denominator become zero. In your function, if x=3 then the denominator becomes zero. So your function has a vertical asymptote at x=3. So we will start our graph by drawing a vertical, dotted line through 3 on the x-axis. (We will learn how to use this line shortly.)
Functions will have horizontal asymptotes if the function values approach a certain number as the x-values get very large (positive or negative). Let's think about your function and what happens for large x-values. (It will help if we understand that if the numerator of a fraction stays the same and the denominator gets bigger. Hopefully you understand that 10/3 > 10/30 > 10/50 > 10/10000, etc. and that these fractions, as the denominator grows, gets closer and closer to zero.
This is what happens to your function. As the x-values keep increasing, the fraction keeps getting closer to zero because the numberator (which doesn't have an x) stays at 1. So your function's graph will have a horizontal asymptote at y = 0 (which is the x-axis). So our horizontal asymptote is the x-axis. The graph of y = 1/(x-3) will get closer and closer to (but never cross!) the x-axis as the x-values get larger and larger, positively or negatively (IOW far to the right and far the left).
At this point we have two asymptotes: x=3 and y=0. But we have not actually started graphing the actual equation! We will start by plotting some points. Given the asymptotes I would suggest the following x-values: 0, 1, 2, 4, 5, and 6. Find the y-values for each of these and plot these 6 points.
Now we will figure out how the graph approaches out asymptotes. Think about an x-value just a little larger (to the right of) 3. Let's try 3.1. If we use this in the function we get y = 1/(3.1-3) = 1/0.1 = 10. If we try even closer to 3 (but still larger than 3) like 3.01, we get y = 1/(3.01-3) = 1/0.01 = 100. I hope you can see that as the x-values approach 3 from the right the y=values become larger and larger (positive) values.
Next think about large positive x-values. We already know that the y-values approach zero. But we also need to recognize that that these small fractions are all positive. So as the graph goes out to the right it will get closer and closer to 0 from above (because above the x-axis is where the numbers are positive).
We are now ready to sketch the graph which is to the right of x=3. We know that just to the right of our vertical asymptote (x=3) the graph will be very high. So we start by placing our pencil at the top of our graph just to the right of the vertical asymptote. Then draw a curve, as smooth as possible through the points we already have for the x-values: 4, 5 and 6. And we finish by continuing the curve to the right making get get closer and closer to the x-axis.
And finally we can repeat this process to the left of the vertical asymptote. We should find that for x-values just below (to the left of) 3 that the y-values become very large negative numbers. And as the x-values become very large negative numbers that the y-values are all very small negative fractions. So now we can place our pencil at the bottom just to the left of the vertical asymptote. Then we draw as smooth a curve as possible through the points we already have for the x-values: 2, 1, and 0 (in that order). And we finish the curve by approaching the x-axis (the horizontal asymptote) from below (where the fractions are negative).
Your graph should like something like the one below. Important: The graph below is not correct! Algebra.com's graphing software does not handle this equation correctly. The vertical asymptote should be dotted, not solid. And the graph should not appear to intersect either of the asymptotes (x=3 and y=0).
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