How do you graph #y= (2x)/(x-1)#?

Answer 1

To graph the equation y = (2x)/(x-1), we can follow these steps:

  1. Determine any restrictions on the domain. In this case, the function is undefined when the denominator (x-1) equals zero. So, x cannot be equal to 1.

  2. Find the y-intercept by substituting x = 0 into the equation: y = (2(0))/(0-1) = 0

  3. Determine the x-intercept by substituting y = 0 into the equation and solving for x: 0 = (2x)/(x-1) 2x = 0 x = 0

  4. Plot the intercepts on the graph.

  5. Analyze the behavior of the function as x approaches positive and negative infinity. As x approaches infinity, y approaches 2. As x approaches negative infinity, y also approaches 2.

  6. Determine the vertical asymptote by finding the values of x that make the denominator equal to zero. In this case, x = 1 is the vertical asymptote.

  7. Sketch the graph, considering the intercepts, asymptote, and the behavior of the function.

The graph of y = (2x)/(x-1) will have a y-intercept at (0, 0), an x-intercept at (0, 0), a vertical asymptote at x = 1, and the function will approach y = 2 as x approaches positive and negative infinity.

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Answer 2

Graph as #y = (2x)/(x-4)#
by establishing a few data points with random values of #x# and noting the asymptotic limits at #y=2# and #x=1#

The asymptotic limit #x=1# should be obvious from the expression (since division by 0 is undefined).
#y=(2x)/(x-1)# is equivalent to #y=2/(1-1/x)# [provided we ignore the special case #x=0#] As #x rarr +-oo# #color(white)("XXXX")##1/x rarr 0# and #color(white)("XXXX")##y=2/(1-1/x) rarr 2/1 = 2# giving the horizontal asymptotic limit.
A few test values for #x#, such as #color(white)("XXXX")##x=-1 rarr y = -1# #color(white)("XXXX")##x=0 rarr y=0# #color(white)("XXXX")##x=2 rarr y = 4# #color(white)("XXXX")##x=3 rarr y= 3#

help give shape to the graph

graph{(2x)/(x-1) [-25.3, 26, -11.27, 14.4]}

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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.

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