Rational functions

Rational Functions,[object Object],By: Katie Thayer,[object Object]

A rational function is any function which can be written as the ratio of two polynomial functions.,[object Object],What is a Rational Function?,[object Object]

Asymptotes,[object Object],Asymptote - a straight line that is the limiting value of a curve,[object Object],The red line on the graph indicates that y=x is the asymptote. ,[object Object]

2 types,[object Object],There are two types of asymptotes. ,[object Object],Vertical Asymptote- x=___, ** CAN’T BE CROSSED**. Can be found be the zeros, or x- intercepts, which are in the denominator.,[object Object],Horizontal Asymptote- y=___, ** Can be crossed*. But there are three different ways you can find a horizontal asymptote.,[object Object]

VA and HA,[object Object],The Vertical Asymptote of out function would be, x=-3 and x=1,[object Object],The Horizontal Asymptote is y=0.,[object Object]

Three types of Horizontal asymptote.,[object Object],Type 1: when the degree of the denominator- degree being the exponential value of x- is larger than the degree of the numerator, then the HA is y=0.,[object Object],Type 2: When the numerator degree and the denominator degree are the same , y=,[object Object],The HA= y= ,[object Object],* When the power of the numerator is bigger than the denominator, the asymptote with be a slant, or oblique. To find the horizontal asymptote of these equations you have to use long division. ,[object Object]

X-intercepts. ,[object Object],To find the x- intercepts you have to take the numerator and set it equal to zero. Going back to the first equation this is what find the x-ints, would look like. The you solve for x by dividing both sides by -5 . So you would get, and since zero can’t be divided , There are no x- intercepts.,[object Object]

Y- intercepts. ,[object Object],To find the y- intercepts you set x = to zero. And it equals 0, so the y- intercept would be at the origin, (0,0).,[object Object]

tables,[object Object],To find possible points of where the line would cross on a graph, you make an x-y tables, of x’s to the left and right of each asymptote.,[object Object]

Domain and Range,[object Object],The domain of a function is the set of all possible x values which will make the function "work" and will output real y-values,[object Object],The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.,[object Object]

The domain in the graph is (-∞,o)U (0,∞)The range would be the same.,[object Object]

Domain and Range cont.,[object Object],The Domain of this function would be,[object Object],D: (-∞,-3] U [-3,1] U [1,∞),[object Object],The Range would be (-∞,∞),[object Object]

Rational functions

Rational functions

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