How do I graph the function: #y=(2x)/(x^21)#?
I like to identify the following things first, when asked to graph a rational function:
 yintercept(s)
 xintercept(s)
 vertical asymptote(s)
 horizontal asymptote(s)

To identify the yintercept(s), ask yourself "what is the value of y when x=0"?
#y = (2(0))/((0)^21)=0/1=0#
yintercept: (0,0) 
To identify the xintercept(s), ask yourself "what is the value of x when y=0"?
For this problem, since we've already identified that the graph goes through (0,0), we have both the xint and yint complete! But in case you didn't realize...
#0=(2x)/(x^21)# means that the numerator of the fraction must = 0
#0=2x#
#0=x#
xintercept: (0,0) 
To identify the vertical asymptotes, we first try and simplify the function as much as possible and then look at where it is undefined
#y=(2x)/(x^21)#
#y=(2x)/((x+1)(x1))#
Undefined when denominator = 0:#(x+1)(x1)=0#
Vertical asymptotes:#x=1, x=1# 
To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
#y=(2x)/(x^21) > y = "huge" / "HUGER" > 0#
Horizontal asymptote:#y=0# Now you might pick a couple additional points to the left/right/between your horizontal asymptotes to get a sense of the graph shape.

Pick a point to the left of the
#x=1# asymptote, ie:#x=2#
#y=(2(2))/((2)^21) = 4/(41) = 4/3# Point 1:#(2, 4/3)# 
Pick a point between the two asymptotes We already have the point (0,0) from above. Point 2:
#(0,0)# 
Pick a point to the right of the
#x=1# asymptote, ie:#x=2#
#y=(2(2))/((2)^21) = 4/(41) = 4/3# Point 3:#(2, 4/3)#
Domain:#(oo,1),(1,1),(1,oo)#
Range:#(oo,oo)#

By signing up, you agree to our Terms of Service and Privacy Policy
To graph the function y=(2x)/(x^21), follow these steps:

Determine the domain of the function by finding the values of x for which the denominator (x^21) is not equal to zero. In this case, x cannot be equal to 1 or 1.

Identify any vertical asymptotes by finding the values of x that make the denominator zero. In this case, x=1 and x=1 are vertical asymptotes.

Determine the behavior of the function as x approaches positive or negative infinity. Divide the leading term of the numerator (2x) by the leading term of the denominator (x^2) to find the horizontal asymptote. In this case, the horizontal asymptote is y=0.

Find the xintercepts by setting y=0 and solving for x. In this case, the xintercept is x=0.

Plot the vertical asymptotes, horizontal asymptote, and xintercept on the coordinate plane.

Choose additional xvalues within the domain and evaluate the function to find corresponding yvalues.

Plot these points on the graph.

Connect the points smoothly to form the graph of the function.
Remember to label the axes and provide a title for the graph.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 How do you graph #f(x)=(2x^32x^2)/(x^39x)# using holes, vertical and horizontal asymptotes, x and y intercepts?
 How do you simplify #(2z^2  11z + 15)/(z^2  9)#?
 What are the asymptote(s) and hole(s), if any, of # f(x) =1/x^21/(1x)+x/(3x)#?
 How do you find the LCM for #y^3y^2, y^4y^2#?
 How do you solve #\frac { 7x  49} { 3x ^ { 2} + 6x  72} + \frac { 1} { x + 6} = \frac { 5} { 3x  12}#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7