Understanding Hyperbolas and Asymptotes

The video tutorial introduces rational functions, focusing on hyperbolas. It explains how to graph hyperbolas by identifying their shape, intercepts, and asymptotes. The tutorial delves into the behavior of horizontal and vertical asymptotes, using tables of values to illustrate these concepts. It also introduces limits and notation to describe asymptotic behavior, emphasizing the differences between horizontal and vertical asymptotes.

What is a rational function primarily composed of?

• Two constants divided by each other

• A constant divided by a polynomial

• A polynomial divided by another polynomial

• A polynomial divided by a constant

Which of the following best describes the shape of a basic hyperbola?

• A circle

• A straight line

• Two separate curves in opposite quadrants

• A single curve in one quadrant

Why is a point for scale necessary when graphing a hyperbola?

• To determine the color of the graph

• To identify the type of asymptote

• To distinguish between different hyperbolas

• To find the intercepts

As x increases, what happens to the value of y in the function y = 1/x?

• y becomes negative

• y remains constant

• y approaches zero

• y approaches infinity

What is the behavior of the function y = 1/x as x approaches zero from the positive side?

• y remains constant

• y approaches infinity

• y approaches zero

• y approaches negative infinity

What is a vertical asymptote associated with in a function?

• A point where the function is undefined

• A point where the function is maximum

• A point where the function is zero

• A point where the function is minimum

How do horizontal asymptotes differ from vertical asymptotes?

• Vertical asymptotes are crossed by the graph, horizontal are not

• Both are always crossed by the graph

• Both are never crossed by the graph

• Horizontal asymptotes are crossed by the graph, vertical are not

What does the notation 'x approaches zero from the positive side' signify?

• x is approaching zero from the left

• x is approaching zero from the right

• x is decreasing

• x is increasing

What is the significance of limits in understanding asymptotes?

• They describe the behavior of the function near asymptotes

• They determine the color of the graph

• They help in finding the intercepts

• They are used to calculate the area under the curve

Which of the following statements is true about asymptotes?

• Vertical asymptotes can be crossed by the graph

• Vertical asymptotes occur at x = 0

• Horizontal asymptotes can be crossed by the graph

• Horizontal asymptotes are always at y = 0