Graph Rational Function By Hand

• MHB
In summary, to graph $f(x) = \frac{2}{(x-3)}$ on the xy-plane by building a table of values, you would need to use x values from -10 to +10 with increments of 1, 20 in total. The graph will be smoother if you use more values, and you should be familiar with transformations to help with graphing.
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:

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

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. :)

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:

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

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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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.

What is a rational function?

A rational function is a mathematical function that can be expressed as the ratio of two polynomial functions. It can be written in the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions and q(x) is not equal to 0.

What is the process for graphing a rational function by hand?

The process for graphing a rational function by hand includes finding the horizontal and vertical asymptotes, plotting the x and y intercepts, and finding the behavior of the function near the asymptotes. Then, using the points and asymptotes, the curve of the function can be sketched.

What are the common mistakes made when graphing rational functions by hand?

Some common mistakes made when graphing rational functions by hand include forgetting to check for vertical asymptotes, graphing the function incorrectly near the asymptotes, and forgetting to label the axes and important points on the graph.

How can I check my hand-drawn graph of a rational function?

To check your hand-drawn graph of a rational function, you can use a graphing calculator or an online graphing tool to compare your results. You can also check your graph by plugging in x-values and comparing the resulting y-values to those on your graph.

Can I use a table of values to graph a rational function by hand?

Yes, a table of values can be used to graph a rational function by hand. By choosing various x-values and calculating the corresponding y-values, you can plot points on a graph and then draw a smooth curve through those points to create the graph of the function.

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