How do you graph #f(x)=(x-2)/(x-4)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Abigail Allen · Accepted Answer

Undefined at #x=4#
See explanation for the rest

#color(blue)("Hole - undefined")#

Hole is where the denominator becomes 0. The function becomes undefined at that point. So for this condition we have #x=4#

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Lets consider the behaviour close to #x=4#

#color(blue)("Vertical Asymptot "->+oo)#

If #x=4+delta4# where #delta4>0# and very small then we have

#(4+delta4-2)/(cancel(4)+delta4-cancel(4)) ->2/(delta4)+1#

#lim_(delta4->color(white)()^+ 0)2/(delta4)+1 color(white)("dd")->color(white)("dd")kcolor(white)("dd")->color(white)("dd")+oo+1=+oo#

#color(blue)("Vertical Asymptot "->-oo)#

If #x=4-delta4# where #delta4>0# and very small then we have

#(4-delta4-2)/(cancel(4)-delta4-cancel(4)) ->(2-delta4)/(-delta4)=-2/(delta4)+1#

#lim_(delta4->color(white)()^+ 0)-2/(delta4)+1color(white)("dd") ->color(white)("dd")kcolor(white)("dd")->color(white)("dd")-oo+1=-oo#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Horizontal Asymptot "->+oo)#

#(x-2)/(x-4)#

As #x>0# becomes increasing greater and greater the influences of the -2 and -4 become less and less significant. This continues until we have

#lim_(x->+oo) (x-2)/(x-4)color(white)("dd")->color(white)("dd")kcolor(white)("dd") ->color(white)("dd")oo/oo=+1#

As #x<0# becomes increasing less and less the influences of the -2 and -4 become less and less significant. This continues until we have

#lim_(x->-oo) (x-2)/(x-4)color(white)("dd")->color(white)("dd")kcolor(white)("dd") ->color(white)("dd")(-oo)/(-oo)=+1#

Isaiah Anthony · Answer

undefined To graph the function f(x) = (x-2)/(x-4), follow these steps:

1. Determine the vertical asymptote: Set the denominator (x-4) equal to zero and solve for x. The vertical asymptote is x = 4.

2. Identify any holes: Simplify the function by canceling out common factors between the numerator and denominator. If any factors cancel out, there will be a hole at that x-value. In this case, there are no common factors to cancel out, so there are no holes.

3. Find the x-intercept: Set the numerator (x-2) equal to zero and solve for x. The x-intercept is x = 2.

4. Find the y-intercept: Substitute x = 0 into the function and solve for y. The y-intercept is (0, -1/2).

5. Determine the horizontal asymptote: Compare the degrees of the numerator and denominator. Since they have the same degree (1), divide the leading coefficients. The horizontal asymptote is y = 1.

6. Plot the points: Plot the x-intercept, y-intercept, and any other desired points on the graph.

7. Draw the graph: Connect the points smoothly, approaching the vertical asymptote at x = 4 and the horizontal asymptote at y = 1.