Graph the Following:
f(x)= (x^3-4x^2+x+6)(x-3)/(x^2-4x+3)
Show All asymptotes, holes, and intercepts and show the table of values you used to graph the function. Pay attention to detail on your graph.
Thanks in advance!
Great, so vertical asymptotes are x=1,3.
Now that cancels with the numerator leaving $\displaystyle \displaystyle f(x)= \frac{x^3-4x^2+x+6}{x-1}$
We need to next make it look a bit nicer, so use polynomial long division on the expression, what do you get?
So now we have
$\displaystyle \displaystyle f(x)= x^2+3x+4 + \frac{10}{x-1}$ which means $\displaystyle \displaystyle x^2+3x+4$ is an oblique asymptote.
Now intercept time.
$\displaystyle \displaystyle f(x)= x^2+3x+4 + \frac{10}{x-1}$
For x-intercepts make $\displaystyle f(x)=0$ and solve.
For y-intercepts make $\displaystyle x=0$ .
Remember that in a rational function $\displaystyle f(x) = 0$ when the function's numerator is 0. Obviously, this is because 0 divided by any number other than 0 is 0 itself.
The shortcut to solving x intercepts in rational functions is to simply set the numerator equal to zero and solve. Try that!
Note that this means that x= 3 is NOT really a "vertical asymptote" as, one either side of x= 3, the graph does not go to infinity or -infinity. Instead, as x approaches 3, f(x) gets closer and closer to (3, 27- 36+ 3+ 6)= (3, 0) but that (3, 0) is not a point on the graph. That's a "hole", not an asyptote. Other than that, Pickslide's advice is excellent!
We need to next make it look a bit nicer, so use polynomial long division on the expression, what do you get?
his can be explained with a formula: (y2 - y1)/(x2 - x1). To find the slope, you pick any two points on the line and find the change in y, and then divide it by the change in x. Example:
1. Problem: The points (1,2) and (3,6) are on a line.
Find the line's slope.
Solution: Plug the given points into
the slope formula.
y2 - y1
m = -------
x2 - x1
6 - 2
m = -----
3 - 1
After simplification, m = 2
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A function is a relation (usually an equation) in which no two ordered pairs have the same x-coordinate when graphed.
One way to tell if a graph is a function is the vertical line test, which says if it is possible for a vertical line to meet a graph more than once, the graph is not a function. The figure below is an example of a function.
Functions are usually denoted by letters such as f or g. If the first coordinate of an ordered pair is represented by x, the second coordinate (the y coordinate) can be represented by f(x). In the figure below, f(1) = -1 and f(3) = 2.
When a function is an equation, the domain is the set of numbers that are replacements for x that give a value for f(x) that is on the graph. Sometimes, certain replacements do not work, such as 0 in the following function: f(x) = 4/x (you cannot divide by 0). In that case, the domain is said to be x <> 0.
There are a couple of special functions whose graphs you should have memorized because they are sometimes hard to graph. They are the absolute value function (below)
and the greatest integer function (below).
The greatest integer function, y = [x] is defined as follows: [x] is the greatest integer that is less than or equal to x.
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