Understanding Graph Transformations of Rational Functions

The video tutorial explains how to match functions with their corresponding graphs by analyzing transformations. It begins with an introduction to rational functions and their asymptotes. The first example involves the function g(x) = 1/(x-4) + 3, highlighting its vertical and horizontal shifts. The tutorial then demonstrates how to express g(x) as a translation of f(x) using function notation. The correct graph is identified by analyzing asymptotes and graph regions. A second example, g(x) = 1/(x+3) - 4, is provided to reinforce the concept of function transformations and graph matching.

What is the basic form of the rational function discussed in the video?

• f(x) = x^2

• f(x) = x + 1

• f(x) = 1/x

• f(x) = 1/(x+1)

What is the vertical asymptote of the function g(x) = 1/(x-4) + 3?

• x = 3

• x = -4

• x = 4

• x = 0

How is the function g(x) = 1/(x-4) + 3 transformed from f(x) = 1/x?

• Shifted left 4 units and up 3 units

• Shifted right 4 units and down 3 units

• Shifted right 4 units and up 3 units

• Shifted left 4 units and down 3 units

What is the horizontal asymptote of the function g(x) = 1/(x-4) + 3?

• y = 4

• y = -3

• y = 3

• y = 0

In the second example, what is the vertical asymptote of g(x) = 1/(x+3) - 4?

• x = 4

• x = 3

• x = -4

• x = -3

How is the function g(x) = 1/(x+3) - 4 transformed from f(x) = 1/x?

• Shifted left 3 units and down 4 units

• Shifted right 3 units and down 4 units

• Shifted left 3 units and up 4 units

• Shifted right 3 units and up 4 units

What is the horizontal asymptote of the function g(x) = 1/(x+3) - 4?

• y = 4

• y = -3

• y = -4

• y = 0

Which graph correctly represents the function g(x) = 1/(x+3) - 4?

• The graph with vertical asymptote x = -4 and horizontal asymptote y = -3

• The graph with vertical asymptote x = 4 and horizontal asymptote y = 3

• The graph with vertical asymptote x = -3 and horizontal asymptote y = -4

• The graph with vertical asymptote x = 3 and horizontal asymptote y = 4

What is the effect of adding a constant to the denominator of a rational function?

• Reflects the graph across the x-axis

• Shifts the graph vertically

• Shifts the graph horizontally

• Changes the slope of the graph

What is the effect of adding a constant to the entire rational function?

• Reflects the graph across the y-axis

• Changes the slope of the graph

• Shifts the graph horizontally

• Shifts the graph vertically