Graphing Functions and Asymptotes

The video tutorial explains how to identify holes and asymptotes in graphs of functions. It begins with an introduction to graphing functions and asymptotes, followed by specific examples such as y = 1/x and y = 1/(x-2). The tutorial distinguishes between holes and asymptotes, using factoring to identify them. It addresses common misconceptions about graphing and provides an example with y = (x^3-x)/(x-1). The lesson concludes with a summary of identifying holes in functions.

What is the asymptote of the function y = 1/x?

• y = 0

• x = 0

• y = 1

• x = 1

Why can't x equal 2 in the function y = 1/(x-2)?

• It would make the numerator zero.

• It would make the denominator zero.

• It would make the function negative.

• It would make the function undefined.

What is the difference between a hole and an asymptote in a graph?

• A hole is where the function approaches infinity, and an asymptote is where the function is undefined.

• Both are points where the function approaches infinity.

• A hole is where the function is undefined, and an asymptote is where the function approaches infinity.

• Both are points where the function is undefined.

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

• The function is undefined.

• There is a hole.

• The function is zero.

• There is an asymptote.

What is the result of factoring the numerator in y = (x^2 - 3x + 2)/(x-1)?

• x + 2

• x - 2

• x - 1

• x + 1

Why can't we simply graph what remains after dividing out a common factor?

• Because it changes the function's asymptotes.

• Because it changes the function's intercepts.

• Because it changes the function's range.

• Because it changes the function's domain.

What is the significance of the denominator in the function y = (x^3 - x)/(x-1)?

• It determines the y-intercepts.

• It determines the x-intercepts.

• It determines the slope.

• It determines the presence of a hole.

What happens to the graph of y = (x^3 - x)/(x-1) at x = 1?

• There is an asymptote.

• There is a hole.

• The graph is undefined.

• The graph is continuous.

What is the shape of the graph of y = x(x+1) after factoring?

• A hyperbola

• A parabola

• A circle

• A line

What must we remember about the original function when graphing y = x(x+1)?

• The original function's domain

• The original function's range

• The original function's asymptotes

• The original function's intercepts