MHB Graph Rational Function By Hand

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To graph the function f(x) = 2/(x - 3) by hand, it's recommended to use a range of x values from -10 to +10 in increments of 1 for a comprehensive view of the graph. There is no strict rule on the number of values to use, but more points yield a smoother graph. Understanding transformations is crucial; the function can be viewed as a transformation of g(x) = 1/x, shifted right by 3 units and vertically stretched. Key features include a vertical asymptote at x = 3 and a horizontal asymptote at y = 0, with the function being positive for x > 3 and negative for x < 3. Graphing by hand remains an important skill despite the availability of calculators, as it enhances comprehension of function behavior.
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Graph $f(x) = \frac{2}{(x - 3)}$ on the xy-plane by building a table of values. 1. How many values of x must I use to graph this function?2. Must I use the same amount of negative values of x as positive values of x to form an even number of points in the form (x, y)?3. Is graphing by hand an important skill to know considering that graphing calculators do the job for us?
 
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Hi RTCNTC,

1) There's no hard rule on how many values you should use. The more values you use the smoother/accurate the graph would be.

2) Again this depends on which domain you want to draw the graph in.

I would choose $x$ values from -10 to +10 with increments of 1 (total of 20 values) to draw this graph, so that I have an understanding of what happens in negative values as well as positive values.
 
Another thing that helps with graphing functions by hand is to look at transformations...for example in this problem we should observe that:

$$f(x)=2g(x-3)$$

where:

$$g(x)=\frac{1}{x}$$

You should be familiar with how the graph of $g$ appears, and we see that $f$ is just moved to the right 3 units and vertically stretched by a factor of 2 relative to $g$. This will give you a good idea of what to expect before you begin constructing your table of points on the given function. :)
 
MarkFL said:
Another thing that helps with graphing functions by hand is to look at transformations...for example in this problem we should observe that:

$$f(x)=2g(x-3)$$

where:

$$g(x)=\frac{1}{x}$$

You should be familiar with how the graph of $g$ appears, and we see that $f$ is just moved to the right 3 units and vertically stretched by a factor of 2 relative to $g$. This will give you a good idea of what to expect before you begin constructing your table of points on the given function. :)

There's an entire chapter dedicated to transformations in Cohen's book but I am not there yet. I honestly think that posting every even number problem from David Cohen's book will take me years to complete one course. I will post the essentials of precalculus from now on by searching online for topics that every precalculus student should know well before stepping into a first semester calculus course.

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Sudharaka said:
Hi RTCNTC,

1) There's no hard rule on how many values you should use. The more values you use the smoother/accurate the graph would be.

2) Again this depends on which domain you want to draw the graph in.

I would choose $x$ values from -10 to +10 with increments of 1 (total of 20 values) to draw this graph, so that I have an understanding of what happens in negative values as well as positive values.

Cool. I thank you for your input. I have decided to post the essentials of precalculus from now on. Keep in mind that this a self-study of a course I took in 1993. I got an A minus in precalculus at Lehman College. Not bad for someone majoring in sociology at the time.
 
Without doing any "calculation" or using a calculator (mine is on the other side of the room and I can't be bothered to walk that far) I would first see that "x- 3" and think "okay, there is a vertical asymptote at x= 3". I would also not that, as x goes to infinity, the numerator stays the same while the denominator gets bigger and bigger so the fraction goes to 0. I would also see that the same thing happens as x goes to negative infinity. The graph gets closer and closer to y= 0 as x goes to positive or negative infinity: y= 0 is a horizontal asymptote. Finally, I see that for x positive the whole fraction is positive while if x is negative, the whole fraction is negative. That is, the graph goes from the top of the graph at x= 3 and curves down to the x-axis to the right side of the graph but goes from 0 on the left up to the top of the graph at x= 3.
 
I will graph it tomorrow.
 
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