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

Question

Genesis Allen · Accepted Answer

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Please read the 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)#

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:

#color(green)("Step 3"#

Generate a data table as follows:

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

#color(green)("Step 4"#

Graph:

Hope you find this useful.

Jameson Allen · Answer

undefined To graph the equation y = (2x + 1)/(x - 5), you can follow these steps:

1. Determine any vertical asymptotes by setting the denominator equal to zero and solving for x. In this case, x - 5 = 0, so x = 5 is a vertical asymptote.

2. Determine any horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (1), there is a horizontal asymptote at y = 0.

3. Find the x-intercept by setting y = 0 and solving for x. In this case, (2x + 1)/(x - 5) = 0, which gives 2x + 1 = 0. Solving for x, we get x = -1/2.

4. Plot the vertical asymptote at x = 5 and the horizontal asymptote at y = 0.

5. Choose some x-values to the left and right of the vertical asymptote (e.g., x = 4 and x = 6) and calculate the corresponding y-values using the equation. Plot these points on the graph.

6. Connect the plotted points with a smooth curve, approaching the vertical asymptote and horizontal asymptote as x approaches infinity or negative infinity.

The resulting graph should show a vertical asymptote at x = 5, a horizontal asymptote at y = 0, and a curve passing through the x-intercept at x = -1/2.