Question

Sketch the asymptotes and graph the function y=4/(x-1)+5​

Answer 1

Answer:

Step-by-step explanation:

The function to be analyzed is:

$y = \frac{4}{x-1}+5$

This function has a vertical and a horizontal asymptote. The vertical asymptote is located where discontinuity exist. That is:

$x = 1$

Besides, the horizontal asymptote coincides with the limit of function, which is:

$\lim_{x \to \pm \infty} \left(\frac{4}{x-1} + 5\right)$

$\lim_{x \to \infty} \frac{4}{x-1} + \lim_{x \to \infty} 5$

$L = 0 + 5$

$L = 5$

The horizontal asymptote is:

$y = 5$

The function and the asymptotes are presented in the image attached below.

Answer 2

orizontal Asymptote:

y

=

0

Vertical Asymptote:

x

=

1

Refer to the graph of

y

=

1

x

when you graph

y

=

4

x

−

1

might help you get some idea of the shape of this function.

graph{4/(x-1) [-10, 10, -5, 5]}

Explanation:

Asymptotes

Find the vertical asymptote of this rational function by setting its denominator to

0

and solving for

x

.

Let

x

−

1

=

0

x

=

1

Which means that there's a vertical asymptote passing through the point

(

1

,

0

)

.

*FYI you can make sure that

x

=

1

does give a vertical asymptote rather than a removable point of discontinuity by evaluating the numerator expression at

x

=

1

. You can confirm the vertical asymptote if the result is a non-zero value. However if you do end up with a zero, you'll need to simplify the function expression, remove the factor in question, for example

(

x

−

1

)

, and repeat those steps. *

You may find the horizontal asymptote (a.k.a "end behavior") by evaluating

lim

x

→

∞

4

x

−

1

and

lim

x

→

−

∞

4

x

−

1

.

If you haven't learned limits yet, you'll still able to find the asymptote by plugging in large values of

x

(e.g., by evaluating the function at

x

=

11

,

x

=

101

, and

x

=

1001

.) You'll likely find that as the value of

x

increase towards positive infinity, the value of

y

getting closer and closer to- but never reaches

0

. So is the case as

x

approaches negative infinity.

By definition , we see that the function has a horizontal asymptote at

y

=

0

Graph

You might have found the expression of

y

=

1

x

, the

x

-reciprocal function similar to that of

y

=

4

x

−

1

. It is possible to graph the latter based on knowledge of the shape of the first one.

Consider what combination of transformations (like stretching and shifting) will convert the first function we are likely familiar with, to the function in question.

We start by converting

y

=

1

x

to

y

=

1

x

−

1

by shifting the graph of the first function to the right by

1

unit. Algebraically, that transformation resembles replacing

x

in the original function with the expression

x

−

1

.



generated with fooplot

Finally we'll vertically stretch the function

y

=

1

x

−

1

by a factor of

4

to obtain the function we're looking for,

y

=

4

x

−

1

. (For rational functions with horizontal asymptotes the stretch would effectively shifts the function outwards.)



generated with footplot