Understanding Rational Functions and Asymptotes

The video tutorial explores the graph of a rational function, f(x) = (2x + 10) / (5x - 15). It begins by identifying intercepts and undefined points, leading to the discovery of vertical asymptotes. The behavior of the function near these asymptotes is analyzed by testing values close to x=3. The concept of horizontal asymptotes is introduced as x approaches infinity, showing how the function approaches a value of 2/5. The tutorial concludes with a summary of how these elements help in sketching the graph of the function.

What is the value of f(x) when x equals zero for the function f(x) = (2x + 10) / (5x - 15)?

• Undefined

• -2/3

• 0

• 2/3

At which x-value is the function f(x) = (2x + 10) / (5x - 15) undefined?

• x = 5

• x = -5

• x = 3

• x = 0

What type of asymptote is present at x = 3 for the function f(x) = (2x + 10) / (5x - 15)?

• No asymptote

• Vertical asymptote

• Point discontinuity

• Horizontal asymptote

What happens to f(x) as x approaches 3 from the left for the function f(x) = (2x + 10) / (5x - 15)?

• Approaches zero

• Approaches negative infinity

• Approaches positive infinity

• Remains constant

What is the behavior of f(x) as x approaches 3 from the right for the function f(x) = (2x + 10) / (5x - 15)?

• Approaches negative infinity

• Approaches zero

• Approaches positive infinity

• Remains constant

As x approaches infinity, what value does f(x) approach for the function f(x) = (2x + 10) / (5x - 15)?

• 0

• 1

• 2/5

• 5/2

What is the horizontal asymptote of the function f(x) = (2x + 10) / (5x - 15)?

• y = 1

• y = 5/2

• y = 2/5

• y = 0

How does the function f(x) = (2x + 10) / (5x - 15) behave as x becomes very negative?

• Approaches 2/5

• Approaches infinity

• Approaches -2/5

• Approaches 0

Which of the following is a step in sketching the graph of a rational function?

• Finding the derivative

• Identifying asymptotes

• Determining the slope

• Calculating the integral

What is the significance of the horizontal asymptote in the graph of a rational function?

• It indicates where the function is undefined

• It shows the end behavior of the function

• It is where the function equals zero

• It determines the slope of the function