Graphing Rational Functions

Summary

This lesson teaches how to graph rational functions by following a systematic approach. Key steps include expressing the function in factored form, simplifying it, identifying x and y-intercepts, and finding vertical and horizontal asymptotes. The instructor also explains how to locate holes and points of intersection with asymptotes. A Cartesian coordinate system is used to plot important details and sketch the graph, ensuring it aligns with theoretical calculations. The process is demonstrated with examples, and viewers are encouraged to practice graphing rational functions.

Takeaways

• 📚 The lesson's objective is to graph rational functions, which requires understanding of intercepts, zeros, asymptotes, and how to solve rational inequalities.
• 🔍 To graph a rational function, first express it in factored form and simplify it to its lowest terms.
• 🔍 Identify any holes in the function by looking for common factors in the numerator and denominator that can be canceled out.
• 📏 Determine the x-intercepts by setting the function equal to zero and solving for x, which involves cross multiplication.
• 📏 Find the y-intercept by substituting x with zero in the function and solving for y.
• 📉 Locate vertical asymptotes by setting the denominator of the function equal to zero and solving for x.
• 📈 Identify horizontal asymptotes by comparing the degrees of the numerator and denominator, and their leading coefficients.
• 🔍 Check for slant (oblique) asymptotes by dividing the numerator by the denominator if the degree of the numerator is one higher than the denominator.
• 📊 Construct a table to analyze the behavior of the function across different regions determined by the asymptotes and intercepts.
• 🖊️ Sketch the graph by plotting the important details such as intercepts, asymptotes, and any holes, and then connect the regions accordingly.
• 🔗 Compare the manually sketched graph with one generated by a graphing application like Geogebra or Desmos to ensure accuracy.

Q & A

What is the objective of the lesson described in the transcript?

The objective of the lesson is to graph rational functions.

What steps are suggested to fully comprehend the lesson on graphing rational functions?

To fully comprehend the lesson, it is suggested to watch videos on intercepts, zeros, asymptotes, wholes, and solving rational inequalities.

How is the function expressed in factored form in the first example?

The function is expressed in factored form as y = (x - 2) / (x + 2), which is already in its simplest form.

What is the x-intercept for the function y = (x - 2) / (x + 2)?

The x-intercept is at the point (2, 0).

What are the vertical and horizontal asymptotes for the function y = (x - 2) / (x + 2)?

The vertical asymptote is x = -2, and the horizontal asymptote is y = 1.

How is the function simplified to its lowest terms in the second example?

The function is simplified to its lowest terms by canceling out the common factor, resulting in y = (x - 2) / (x + 3).

What is the hole in the function y = (x - 2) / (x + 3), and where is it located?

The hole is at the point (-5, 7/2).

What is the y-intercept for the function y = (x - 2) / (x + 3)?

The y-intercept is at the point (0, -2/3).

How many regions are determined when graphing the function y = (x - 2) / (x + 3)?

There are three regions determined when graphing the function y = (x - 2) / (x + 3).

What is the technique used to determine the sign of the function's value in different regions?

The technique used is to count the number of negative factors in the expression for each region, with an even count resulting in a positive sign and an odd count resulting in a negative sign.

How does the graph approach the asymptotes in the regions where there is no point of intersection?

The graph approaches but does not cross the asymptotes, getting closer as it moves towards the asymptotes.