Sometimes graphing rational functions can feel a little "irrational." Starting with the basics, learners work their way through the pieces of these graphs and finish off with an application question. 

Continuing with the study of graphing hyperbolas, Sal shows an example of graphing a parabola that is centered at a point off the origin. By looking at the equation, first without shifting it, he helps listeners develop an intuitive...

Here is an example in this video of graphing a hyperbola centered at the origin. First, Sal shows solving for the equations of the asymptotes and then shows how to determine the direction of the hyperbola. He takes the approach of not...

In this video, Sal starts with the equation of a hyperbola by first comparing it to the equation of the circle, and ellipse. He then shows how to find the equation of the asymptotes by solving for y. He also shows a few intuitive ways to...

Extending his asymptotes lesson from the previous video, Sal shows viewers another example of graphing rational functions. This video is shorter than the previous one and could be shown in a single class session.

They say it runs in the family. Small groups use a card sorting activity to group graphs of functions into families. After grouping by families, the class develops conclusions for parent functions and create a graphic organizer.

Calculating average speeds is not as simple as finding an average. The activity introduces class members to rational functions by presenting a problem about finding an average speed for the rest of a trip. Pupils develop an equation to...

Test the limits of one's knowledge on continuity. The last of 18 installments in Unit 1 - Limits and Continuity is an end-of-unit review learning exercise. Individuals identify and classify discontinuities, investigate continuity at a...

Learning about end behavior means it's close to the end of the unit. The 16th of 18 lessons in Unit 1 - Limits and Continuity has scholars watch a video to learn how limits at infinity can be used to determine horizontal asymptotes and...

There's nothing sketchy about a helpful graphing resource. An informative video shows how to sketch graphs of functions. Scholars first apply derivatives to determine increasing/decreasing behavior, concavity, and asymptotes. They then...

Learn about slant asymptotes directly—not obliquely. Viewers of a short video discover how to determine a slant or oblique asymptote for a rational function. They see how to apply polynomial division to achieve this goal.

Is a vertical asymptote required to find the horizontal counterpart? By watching a video, scholars learn how to determine the horizontal asymptote of the graph of a rational function. They recognize that the degrees of the leading terms...

Differentiate between calculus concepts. The beginning of a helpful calculus resource begins with a short review of functions and moves right into limits to define derivatives. The six-chapter resource continues with applications of...

Do not cross the line while graphing. Provided with several coordinate axes along with asymptotes, pupils determine two functions that will fit the given restrictions. Scholars then determine other geometrical relationships of asymptotes...

Where do those asymptotes come from? Learners graph, simplify, and solve rational functions in the fifth module of a 10-part series. Beginning with graphing, pupils determine the key characteristics of the graphs including an in-depth...

Shift the function and transform the key features of the graph. By translating the graph of the rational function, class members find out how the key features alter. Pupils determine the domain, range, asymptotes, and intervals of...

There are an infinite number of reasons to use the resource. Scholars drag vertical and horizontal lines to the graph of a rational function to identify all asymptotes. They investigate the connection between asymptotes and limits to...

Limits can provide some valuable information about graphs. A slider interactive lets learners see the behavior of a graph around asymptotes. They investigate relationships between limits and asymptotes.

Rational functions become less mysterious when you know about limits. Individuals use an interactive to move a rational function on a coordinate plane and to investigate function values for certain x-values. They see how the limit...

Patterns can be shifty! Find the pattern when shifting the graph of tangent. Pupils move the graph of tangent to different locations on the coordinate plane. They observe what happens to the function and its vertical asymptotes before...

Examine the connection between rational functions and their graphs. Individuals use an online manipulative to sort equations with horizontal and oblique asymptotes. They focus on the degree of the numerator and denominator.

Play with the graph of a rational function to discover the asymptote patterns. Young scholars use the interactive lesson to discover the relationship between the asymptotes and the function. As they manipulate the function, the graph...