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If r is a zero of Even Multiplicity, then the graph touches the x-axis and turns around at r.
The Domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. Concavity: We examine the intervals along the x-axis where the curve is either concave up or down. We’ll skip the table of values for this problem, and just show the graphs of the three functions on the same coordinate axes. Notice that we didn’t pick as many values for our table this time, because by now we have a pretty good idea what happens near the vertical asymptote. We can see that this function will have no vertical asymptotes because the denominator of the fraction will never be zero. We can see that the beginning terms in the numerator and denominator are much bigger than the other terms in each expression. There is no horizontal asymptote because the power of the numerator is larger than the power of the denominator.
So, we start plotting the function by drawing the vertical and horizontal asymptotes on the graph. SUMMARY OF HOW TO FIND ASYMPTOTES Vertical Asymptotes are the values that are NOT in the domain. Finding Asymptotes VERTICAL ASYMPTOTES There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0 Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be. If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote.

If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at: y = leading coefficient of top leading coefficient of bottom degree of bottom = 2 HORIZONTAL ASYMPTOTES degree of top = 2 The leading coefficient is the number in front of the highest powered x term. If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one.
RATIONAL FUNCTIONS A rational function is a function of the form: where p and q are polynomials. PARENT FUNCTIONS Constant Function Linear Absolute Value Quadratic Cubic Square Root Rational.
A rational function is the quotient of two polynomials Rational Functions: A rational function has the form where P(x) and Q(x) are polynomials. Vertical Asymptotes (1) Vertical asymptotes exist when the rational function is in lowest terms and its denominator polynomial can be equal to 0 for some. Graphing Rational Functions Example #7 End ShowEnd Show Slide #1 NextNext We want to graph this rational function showing all relevant characteristics.
If a polynomial equation is of degree n, then counting multiple root separately, the equation has n roots.
If a + bi is a root of a polynomial equation with real coefficients(b?0), then the imaginary numbers a-bi is a root.
We determine the intervals along the x-axis where the curve increases and decreases by examining the slopes of the tangent lines. Graphs of rational functions are very distinctive, because they get closer and closer to certain values but never reach those values. Notice that for larger constants of proportionality, the curve decreases at a slower rate than for smaller constants of proportionality.

They are not part of the function; instead, they show values that the function approaches, but never gets to. One way to find the horizontal asymptote of a rational function is to ignore all terms in the numerator and denominator except for the highest powers.
Because our function has a lot of detail we must make sure that we pick enough values for our table to determine the behavior of the function accurately. In addition, resistors are used in an electrical circuit to control the amount of current flowing through a circuit and to regulate voltage levels.
If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. X-intercepts of Rational Function To find the x-int of Rational Functions, set the numerator equal to zero and solve for x.
3.7 Notes Unlike polynomial functions which are continuous, rational functions have discontinuities. One important reason to do this is to prevent sensitive electrical components from burning out due to too much current or too high a voltage level. We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.

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