Rational Functions and Asymptotes
Rational Functions are function rules that equal the quotient of two polynomials. This means that x will be in the denominator. The parent function is f(x) = 1/x and the graph is called a hyperbola.
Notice in the graph and table to the right, the x values will approach 0, from both the positive direction and negative direction, but will never equal 0. This is because dividing by 0 is undefined. The y values will do the same. They will approach 0 in both directions but never actually equal 0. These "holes" in the graph are called asymptotes. The vertical asymptote is the value of x that results in an undefined y. The vertical asymptote for the parent function f(x) = 1/x is simply x = 0, because f(x) = 1/0 is not possible. The horizontal asymptote will be the y value that causes the x value to be undefined. The horizontal asymptote for the parent function is y = 0. There are no possible x values that will cause y to equal 0 (0 = 1/x). See the work shown below. Finding the vertical and horizontal asymptotes of a rational function can be simple if it is written in a certain form. Watch the video below and take notes. After the video complete the practice worksheet that follows by clicking on the link. Check your answer after you have completed the worksheet.
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Parent Function
Vertical Asymptote (x = 0) and Horizontal Asymptote (y = 0)
Vertical Asymptote (x = -3) and Horizontal Asymptote (y = -5)
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Domain, Range, Transformations of Rational Functions
Rational functions can be expressed in h, k form just like quadratic functions and radical functions. In the video above you where shown how to find the vertical and horizontal asymptotes. The way those function rules were written was h, k form. You will now use this form to identify not only the asymptotes, but the domain and range, as well as all transformations.
Domain and RangeRemember that domain is all the possible x values in a function and the range is all the possible y values in a function. Since there are "holes in the graph, the domain and range will be written differently. The video shows the set notation, and the example below explains the interval notation.
Watch the video linked below |
TransformationsJust as before, the h and k will translate the parent function horizontally and vertically. The a value will stretch or compress the function vertically, and reflect it over the x axis.
Copy the h,k form to the right in your notes, then watch the video linked below. |
Complete the worksheet linked below, then check your answers.
Systems Involving Rational Functions
Remember that systems of equations can involve any combination of function rules. You will use the same system rules you always have, the only difference is the way you solve the resulting equation.
Try and work the examples below given your previous knowledge of systems. Click on the solution link to check your work. Also remember the solutions are where the graphs intersect. You can view the graphs of this example by clicking the graph link.
You can also use the graphing calculator to check your answers. Make sure both equations are solved for y and graph them. Hit 2nd, Trace, #5 (intersect). Move the curser to the point(s) of intersection and hit enter 3 times.
Try and work the examples below given your previous knowledge of systems. Click on the solution link to check your work. Also remember the solutions are where the graphs intersect. You can view the graphs of this example by clicking the graph link.
You can also use the graphing calculator to check your answers. Make sure both equations are solved for y and graph them. Hit 2nd, Trace, #5 (intersect). Move the curser to the point(s) of intersection and hit enter 3 times.