ASYMPTOTE HELP
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ASYMPTOTE HELP

Sketch the graph of f(x)=2^-x+3. Indicate the characteristic points and asymptote on the graph

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ASYMPTOTE HELP

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Apr 21, 2011
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by: Staff


The question:

Sketch the graph of f(x)=2^-x+3. Indicate the characteristic points and asymptote on the graph


The answer:


f(x)=2^-x + 3

REWRITE the function AS THE QUOTIENT OF TWO POLYNOMIALS (combine the fractions 1/2^x and 3/1)

f(x)= (1/2^x) + 3 * (2^x /2^x)

f(x)= 1/(2^x) + 3 * 2^x /(2^x)

f(x)= (1 + 3 * 2^x) /(2^x)

f(x)= (3 * 2^x + 1) /(2^x)


Vertical asymptotes occur where a particular value of x will cause the denominator = 0

2^x = 0, solve for x

2^x = undefined, there is no value of x which will cause the denominator to = 0

Therefore, there are NO VERTICAL asymptotes.


Horizontal asymptotes

Since the degree of the numerator and the denominator are the same (the degree is x), this function DOES HAVE a non-zero (not equal to the x axis) HORIZONTAL ASYMPTOTE.

The horizontal asymptote can be found by dividing the first term of the numerator by the first term of the denominator:

f(x)= (3 * 2^x + 1) /(2^x)

horizontal asymptote = 3 * 2^x/2^x = 3

the horizontal asymptote is: f(x) = 3


Slant asymptotes

Since the degree of the numerator and the denominator are the same (the degree is x), this function DOES NOT HAVE a SLANT ASYMPTOTE.


table of values for the function

x, f(x)= (3 * 2^x + 1) /(2^x)


-8 259
-7 131
-6 67
-5 35
-4 19
-3 11
-2 7
-1 5
0 4
1 3.5
2 3.25
3 3.125
4 3.0625
5 3.03125
6 3.015625
7 3.007813


The function can be graphed by plotting the table of values [x & f(x)]

(click link to view graph, use the Backspace key to return to this page):

http://www.solving-math-problems.com/images/function-asymptote-graph.png


The final answer is:

y intercept: (0, 4)
vertical asymptotes: none
horizontal asymptote: y = 3
slant asymptote: none


Thanks for writing.


Staff
www.solving-math-problems.com

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