Identifying Holes in Graphs: Investigating Functions

This video tutorial explains how to identify holes in functions by examining their graphs. It covers the concept of asymptotes and how they indicate where a function is undefined. The tutorial differentiates between holes and asymptotes, showing how to factor and simplify functions to identify these features. It addresses common misunderstandings in graphing functions and provides an example of factoring and graphing a function with a hole. The lesson concludes with a summary of the key points covered.

What happens to the function y = 1 / X when X equals 0?

• The function equals zero

• The function is defined and equals 1

• The function is undefined, creating an asymptote

• The function has a hole at X = 0

In the function y = (X^2 - 3X + 2) / (X - 1), what occurs at X = 1?

• An asymptote

• A hole

• A maximum point

• A minimum point

Why does the function y = (X^2 - 3X - 4) / (X - 1) have an asymptote at X = 1?

• Because X = 1 is a root of the numerator

• Because the denominator cannot be canceled out

• Because X = 1 is a root of the denominator

• Because the numerator and denominator are equal

What is the result of canceling common factors in the function y = (X^2 - 3X + 2) / (X - 1)?

• The function becomes a constant

• The function has a hole

• The function has an asymptote

• The function becomes undefined

What is a common misunderstanding when graphing functions after canceling common factors?

• That the function can be graphed without considering the original denominator

• That the function has multiple asymptotes

• That the function has no intercepts

• That the function becomes a straight line

In the function y = (X^3 - X) / (X - 1), what happens at X = 1 after factoring?

• The function has a hole

• The function has an asymptote

• The function is continuous

• The function has a maximum

What shape does the graph of y = X(X + 1) take after canceling the common factor in y = (X^3 - X) / (X - 1)?

• A hyperbola

• A circle

• A parabola

• A straight line