Graphing Rational Functions Using Transformations With Vertical and Horizontal Asymptotes

Summary

This video focuses on graphing rational functions through transformations, using examples like 1/x and its reflections, as well as 1/x squared. The presenter explains vertical and horizontal asymptotes, domain, and range for each function, highlighting the symmetry of odd and even functions. Transformations such as shifts and reflections are demonstrated with various rational functions, emphasizing the importance of identifying asymptotes when determining domain and range. The video serves as a comprehensive guide to understanding the characteristics of rational functions and their graphical representations.

Takeaways

• 😀 Rational functions can be graphed using transformations of basic functions, like 1/x.
• 📏 The vertical asymptote of a rational function occurs where the denominator equals zero.
• 🌊 The horizontal asymptote of a rational function is determined by the behavior of the function as x approaches infinity.
• 🔄 Reflecting a graph over the x-axis or y-axis can alter the function's sign and shape.
• 🔍 The domain of a rational function includes all x-values except for those where the function is undefined (i.e., vertical asymptotes).
• 🔼 The range of a rational function excludes the horizontal asymptote, as the graph approaches but never touches it.
• 🟰 Odd functions are symmetric about the origin, while even functions are symmetric about the y-axis.
• ➡️ Horizontal shifts in a function occur based on the values added or subtracted from x in the denominator.
• ⬆️ Vertical shifts result from adding or subtracting constants outside the function.
• 🗺️ Understanding how to determine the domain and range is crucial for sketching accurate graphs of rational functions.

Q & A

What are vertical and horizontal asymptotes?

Vertical asymptotes occur where the denominator of a rational function is zero, while horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity.

How do you find the vertical asymptote for the function 1/x?

The vertical asymptote for the function 1/x is found by setting the denominator equal to zero, which gives x = 0.

What is the horizontal asymptote of the function 1/x?

The horizontal asymptote of the function 1/x is the x-axis, represented by the equation y = 0.

How does the graph of -1/x differ from 1/x?

The graph of -1/x is a reflection of the graph of 1/x across the origin, resulting in it existing in quadrants II and IV instead of I and III.

What does it mean for a function to be even or odd?

An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. For example, 1/x is an odd function, and 1/x^2 is an even function.

How does the transformation affect the function 1/(x+2)?

The transformation 1/(x+2) shifts the graph of 1/x two units to the left, resulting in a vertical asymptote at x = -2.

What is the domain of the function 1/x?

The domain of the function 1/x is all real numbers except zero, which can be expressed as (-∞, 0) ∪ (0, ∞).

What is the range of the function 1/x squared?

The range of the function 1/x squared is (0, ∞) because it is always positive.

What happens to the range of a function when a constant is added outside it?

When a constant is added outside a function, it shifts the horizontal asymptote up or down. For example, 1/x + 2 has a horizontal asymptote at y = 2.

How do you determine the domain and range for the function 3 - 1/(x+2)?

The domain is (-∞, -2) ∪ (-2, ∞) excluding the vertical asymptote at x = -2. The range is (-∞, 3) ∪ (3, ∞) excluding the horizontal asymptote at y = 3.