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Graph and Asymptotes of Rational Functions: Domain, Range, Intercepts, and Asymptotes, Cheat Sheet of Mathematics

Mathematics

The graph and properties of rational functions, including the domain, range, intercepts, and asymptotes. The domain of a rational function is found by setting the denominator equal to zero, while the range is determined by identifying points unreachable through input values. Intercepts are located by setting x or y equal to zero and solving for the corresponding variable. Zeros are the same as the numerator zeros, and vertical asymptotes are found by setting the denominator equal to zero. Horizontal asymptotes depend on the degrees of the numerator and denominator.

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2020/2021

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Graph and Properties of Rational Function
Domain of Rational Function
The domain of a rational function consists of all the real numbersxexcept those for which the
denominator is0. To find thesexvalues to be excluded from the domain of a rational function,
equate the denominator to zero and solve forx.
Range of Rational Function
To find the range of a rational function, we need to identify any point that cannot be achieved
from any input; these can generally be found by considering the limits of the function as the
magnitude of the inputs get very large.
Finding Intercepts of Rational Fractions
Intercepts are the points at which a graph crosses either the x or y axis, and they are very useful
in sketching functions.
To find the y-intercept(s) (the point where the graph crosses the y-axis), substitute in 0 for x and
solve for y or f(x).
To find the x-intercept(s) (the point where the graph crosses the x-axis also known as zeros),
substitute in 0 for y and solve for x.
Zeros of Rational Function
When a rational function is equal to zero (that is, its output is equal to zero) then its
NUMERATOR is equal to zero. So, to find the zeros of a rational function we simply find the
zeros of the NUMERATOR.
Asymptotes - A line is an asymptote if the distance between the curve and the line approaches
zero as we move out farther and farther on the line. We can say that the asymptote 'models' the
behavior of the curve.
Vertical Asymptote - When a vertical line is an asymptote then the graph gets closer and closer
to the vertical line. The graph becomes vertical so the vertical line is a model of what the graph
looks like as the graph gets closer to the line.We find the vertical asymptotes by setting the
DENOMINATOR of the function equal to zero.
Horizontal asymptotesare horizontal lines the graph approaches. To find horizontal
asymptotes:
If the degree (the largest exponent) of the denominator isbigger thanthe degree of the
numerator, the horizontal asymptote is the x-axis (y = 0).
If the degree of the numerator isbigger thanthe denominator, there is no horizontal
asymptote.
If the degrees of the numerator and denominator are thesame, the horizontal asymptote
equals the leading coefficient (the coefficient of the largest exponent) of the numerator
divided by the leading coefficient of the denominator

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Graph and Properties of Rational Function Domain of Rational Function The domain of a rational function consists of all the real numbers x except those for which the denominator is 0. To find these x values to be excluded from the domain of a rational function, equate the denominator to zero and solve for x. Range of Rational Function To find the range of a rational function, we need to identify any point that cannot be achieved from any input; these can generally be found by considering the limits of the function as the magnitude of the inputs get very large. Finding Intercepts of Rational Fractions Intercepts are the points at which a graph crosses either the x or y axis, and they are very useful in sketching functions. To find the y-intercept(s) (the point where the graph crosses the y-axis), substitute in 0 for x and solve for y or f(x). To find the x-intercept(s) (the point where the graph crosses the x-axis also known as zeros), substitute in 0 for y and solve for x. Zeros of Rational Function When a rational function is equal to zero (that is, its output is equal to zero) then its NUMERATOR is equal to zero. So, to find the zeros of a rational function we simply find the zeros of the NUMERATOR. Asymptotes - A line is an asymptote if the distance between the curve and the line approaches zero as we move out farther and farther on the line. We can say that the asymptote 'models' the behavior of the curve. Vertical Asymptote - When a vertical line is an asymptote then the graph gets closer and closer to the vertical line. The graph becomes vertical so the vertical line is a model of what the graph looks like as the graph gets closer to the line.We find the vertical asymptotes by setting the DENOMINATOR of the function equal to zero. Horizontal asymptotes are horizontal lines the graph approaches. To find horizontal asymptotes:  If the degree (the largest exponent) of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis (y = 0).  If the degree of the numerator is bigger than the denominator, there is no horizontal asymptote.  If the degrees of the numerator and denominator are the same , the horizontal asymptote equals the leading coefficient (the coefficient of the largest exponent) of the numerator divided by the leading coefficient of the denominator