How do you graph #y =(2x + 1)/(x-5)#?

1 Answer
Aug 2, 2018

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Please read the explanation.

Explanation:

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Given:

Rational function : #color(red)(y=f(x)=(2x+1)/(x-5)#

#color(blue)("How do we graph a rational function ?"#

#color(green)("Step 1"#

Find both #color(red)(x and y# intercepts.

x-intercept:

Set #color(red)(y=0#

#rArr (2x+1)/(x-5)=0#

#rArr (2x+1)=0#

#rArr 2x=-1#

#rArr x=(-1/2)#

#color(red)("x-intercept": (-1/2,0)#

y-intercept:

Set #color(red)(x=0#

#rArr y=(2x+1)/(x-5)#

#rArr y=(2(0)+1)/((0)-5)#

#rArr y = 1/(-5)#

#color(red)("y-intercept": (0,-1/5)#

enter image source here

Both x and y intercepts are plotted in the graph above.

#color(green)("Step 2"#

Find Horizontal and Vertical Asymptotes

Vertical asymptotes are generated by the ZEROS of the denominator.

Horizontal asymptoes describe the behavior of the graph as the input values get larger or smaller.

Vertical asymptote:

Figure out what makes the denominator equal to zero.

Make sure that the numerator does not become zero for the same value.

#rArr (x-5)=0#

#rArr x=5#

Vertical asymptote is at #color(red)(x=5#

Horizontal asymptote:

Numerator = #(2x + 1)#

Highest degree in both numerator and denominator is #color(red)(1#

#rArr 2/1=2#

Horizontal asymptote is at #color(red)(y=2#

You can view the asymptotes in the graph below:

enter image source here

#color(green)("Step 3"#

Generate a data table as follows:

enter image source here

Consider the columns (first and the last): #color(blue)(x and y#:

enter image source here

#color(green)("Step 4"#

Graph:

enter image source here

Hope you find this useful.