Graph Rational Function By Hand

[mathdad]

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?

 

[Sudharaka]

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.

 

[MarkFL]

Another thing that helps with graphing functions by hand is to look at transformations...for example in this problem we should observe that:

\(\displaystyle f(x)=2g(x-3)\)

where:

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

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. :)

 

[mathdad]

Another thing that helps with graphing functions by hand is to look at transformations...for example in this problem we should observe that:

\(\displaystyle f(x)=2g(x-3)\)

where:

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

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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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.

 

[HallsofIvy]

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.

 

[mathdad]

I will graph it tomorrow.