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.

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 instructional activity. Individuals identify and classify discontinuities, investigate continuity at...

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...

Let's see what the class learned! The last installment of the 23-part Pre-calculus module consists of nine assessment questions designed to measure a specific portion of a standard in the unit. Learners demonstrate their understanding of...

Get close to your favorite line. Scholars use end behavior to help find horizontal asymptotes. With the understanding of domains of rational functions, learners find vertical asymptotes and then use graphing calculators to verify the...

Put everything together in a picture. Scholars take what they learned in the previous three lessons in the 15th segment of a 23-part unit to graph rational functions. They find the end behavior, the horizontal and vertical asymptotes if...

Escape to investigate hyperbolas. Pupils take a look at what happens to the elliptical orbital path of a satellite that exceeds escape velocity as the opener to the eighth instructional activity in a unit of 23. Scholars analyze basic...

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...

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.

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.