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Step-by-Step Guidance: Rational Functions and Their Graphs (College Algebra)

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Q1. Find the domain:

Background

Topic: Domain of Rational Functions

This question tests your understanding of how to determine the domain of a rational function by identifying values that make the denominator zero.

Key Terms and Formulas:

  • Domain: The set of all real numbers for which the function is defined.

  • Rational Function: A function of the form , where and are polynomials and .

Step-by-Step Guidance

  1. Identify the denominator of the function: .

  2. Set the denominator equal to zero to find values that are not in the domain: .

  3. Solve for to find the excluded value(s).

  4. Express the domain as all real numbers except the value(s) that make the denominator zero.

Try solving on your own before revealing the answer!

Q2. Find the domain:

Background

Topic: Domain of Rational Functions

This question asks you to determine the domain by finding all -values that make the denominator zero and excluding them from the domain.

Key Terms and Formulas:

  • Domain: The set of all real numbers except those that make the denominator zero.

  • Factoring: You may need to factor the denominator to find all values that make it zero.

Step-by-Step Guidance

  1. Identify the denominator: .

  2. Set the denominator equal to zero: .

  3. Factor the denominator to solve for .

  4. List all -values that make the denominator zero, and exclude them from the domain.

Try solving on your own before revealing the answer!

Q3. For , determine the following limits:

  • As , f(x) -- >

  • As , ______

  • As , ______

  • As , ______

Background

Topic: Limits and Asymptotic Behavior of Rational Functions

This question tests your understanding of how rational functions behave near vertical asymptotes and at infinity.

Key Terms and Formulas:

  • Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

  • Horizontal Asymptote: Describes the end behavior as approaches infinity or negative infinity.

  • Arrow Notation: means approaching from the right; means from the left.

Step-by-Step Guidance

  1. Identify the vertical asymptote by setting the denominator and solving for .

  2. Analyze the sign of the numerator and denominator as approaches $2x \to 2^+x \to 2^-$).

  3. For and , compare the degrees of the numerator and denominator to determine the horizontal asymptote.

  4. Use the leading coefficients to find the value the function approaches as becomes very large or very negative.

Try solving on your own before revealing the answer!

Q4. Find the vertical asymptotes and holes:

Background

Topic: Vertical Asymptotes and Holes in Rational Functions

This question tests your ability to identify vertical asymptotes and holes by analyzing the factors of the numerator and denominator.

Key Terms and Formulas:

  • Vertical Asymptote: Occurs at if the denominator is zero and the numerator is not zero at $x = a$.

  • Hole: Occurs at if both the numerator and denominator are zero at $x = a$ (i.e., they share a common factor).

Step-by-Step Guidance

  1. Factor the numerator and denominator if possible.

  2. Set the denominator equal to zero and solve for to find potential vertical asymptotes or holes.

  3. Check if the numerator is also zero at those -values to determine if there is a hole or a vertical asymptote.

  4. List the -values where vertical asymptotes and holes occur.

Try solving on your own before revealing the answer!

Q5. Find the vertical asymptotes and holes:

Background

Topic: Vertical Asymptotes and Holes in Rational Functions

This question asks you to factor both the numerator and denominator to identify any common factors (which would indicate a hole) and to find vertical asymptotes.

Key Terms and Formulas:

  • Factoring: can be factored as .

  • Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

  • Hole: Occurs where both numerator and denominator are zero (common factor).

Step-by-Step Guidance

  1. Factor the numerator: .

  2. Identify any common factors between numerator and denominator.

  3. Set the denominator equal to zero and solve for .

  4. Determine if the -value is a vertical asymptote or a hole based on the presence of common factors.

Try solving on your own before revealing the answer!

Q6. Find the vertical asymptotes and holes:

Background

Topic: Vertical Asymptotes and Holes in Rational Functions

This question requires you to factor the numerator and analyze the denominator to find vertical asymptotes and holes.

Key Terms and Formulas:

  • Factoring: .

  • Vertical Asymptote: Occurs where the denominator is zero and the numerator is not zero.

  • Hole: Occurs where both numerator and denominator are zero (common factor).

Step-by-Step Guidance

  1. Factor the numerator: .

  2. Set the denominator equal to zero: .

  3. Check if is a factor in the numerator to determine if there is a hole.

  4. List the -values for vertical asymptotes and holes.

Try solving on your own before revealing the answer!

Q7. The graph of

Find the equation of its slant asymptote.

Background

Topic: Slant (Oblique) Asymptotes of Rational Functions

This question tests your ability to find the slant asymptote by dividing the numerator by the denominator when the degree of the numerator is one more than the denominator.

Key Terms and Formulas:

  • Slant Asymptote: Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

  • Long Division: Divide the numerator by the denominator and ignore the remainder.

Step-by-Step Guidance

  1. Set up the long division: divide by .

  2. Perform the division to find the quotient (ignore the remainder).

  3. The equation of the slant asymptote is (the quotient from the division).

Try solving on your own before revealing the answer!

Q8. Find the horizontal, slant, and non-linear asymptotes of

Background

Topic: Asymptotes of Rational Functions

This question asks you to determine the type of asymptote by comparing the degrees of the numerator and denominator, and to find the equation of the asymptote if it exists.

Key Terms and Formulas:

  • Horizontal Asymptote: If degree numerator < degree denominator, ; if equal, .

  • Slant Asymptote: If degree numerator is one more than denominator, divide numerator by denominator.

Step-by-Step Guidance

  1. Compare the degrees of the numerator () and denominator ().

  2. Since the degree of the numerator is one more than the denominator, set up the division divided by .

  3. Perform the division to find the slant asymptote (ignore the remainder).

  4. State whether there is a horizontal asymptote (there is not if degree numerator > denominator).

Try solving on your own before revealing the answer!

Q9. Find the horizontal, slant, and non-linear asymptotes of

Background

Topic: Asymptotes of Rational Functions

This question is similar to the previous one, focusing on identifying and finding the equations of asymptotes.

Key Terms and Formulas:

  • Same as above: compare degrees, perform division if needed.

Step-by-Step Guidance

  1. Compare the degrees of the numerator and denominator.

  2. Set up the division divided by .

  3. Perform the division to find the slant asymptote.

  4. State whether a horizontal asymptote exists.

Try solving on your own before revealing the answer!

Q10. Find the horizontal, slant, and non-linear asymptotes of

Background

Topic: Asymptotes of Rational Functions

This question involves a numerator with degree two higher than the denominator, which may result in a non-linear (parabolic) asymptote.

Key Terms and Formulas:

  • Non-linear Asymptote: If the degree of the numerator is more than one higher than the denominator, divide to find the polynomial asymptote.

Step-by-Step Guidance

  1. Compare the degrees: numerator is degree 3, denominator is degree 1.

  2. Set up the division divided by .

  3. Perform the division to find the polynomial (non-linear) asymptote.

  4. Write the equation of the asymptote using the quotient (ignore the remainder).

Try solving on your own before revealing the answer!

Q11. Find the horizontal, slant, and non-linear asymptotes of

Background

Topic: Asymptotes of Rational Functions

This question involves a numerator with degree much higher than the denominator, so you may get a non-linear asymptote.

Key Terms and Formulas:

  • Same as above: perform polynomial division to find the asymptote.

Step-by-Step Guidance

  1. Compare the degrees: numerator is degree 4, denominator is degree 1.

  2. Set up the division divided by .

  3. Perform the division to find the polynomial asymptote.

  4. Write the equation of the asymptote using the quotient (ignore the remainder).

Try solving on your own before revealing the answer!

Q12. Graph:

Background

Topic: Graphing Rational Functions

This question asks you to graph a rational function by finding intercepts, asymptotes, and analyzing the function's behavior.

Key Terms and Formulas:

  • Y-intercept: Set and solve for .

  • X-intercepts: Set the numerator equal to zero and solve for .

  • Vertical Asymptotes: Set the denominator equal to zero and solve for .

  • Horizontal Asymptote: Compare degrees of numerator and denominator.

Step-by-Step Guidance

  1. Find the y-intercept by evaluating .

  2. Find the x-intercepts by setting and solving for .

  3. Find the vertical asymptotes by setting and solving for .

  4. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator.

  5. Plot these features and sketch the graph accordingly.

Try solving on your own before revealing the answer!

Q13. Graph:

Background

Topic: Graphing Rational Functions

This question involves factoring both numerator and denominator, finding intercepts, asymptotes, and sketching the graph.

Key Terms and Formulas:

  • Same as above: find intercepts, asymptotes, and factor where possible.

Step-by-Step Guidance

  1. Factor the numerator and denominator.

  2. Find the y-intercept by evaluating .

  3. Find the x-intercepts by setting the numerator equal to zero.

  4. Find the vertical asymptotes by setting the denominator equal to zero.

  5. Determine the horizontal asymptote by comparing degrees.

Try solving on your own before revealing the answer!

Q14. Graph:

Background

Topic: Graphing Rational Functions

This question asks you to analyze and graph a rational function with a quadratic numerator and linear denominator.

Key Terms and Formulas:

  • Same as above: find intercepts, asymptotes, and analyze behavior.

Step-by-Step Guidance

  1. Find the y-intercept by evaluating .

  2. Find the x-intercepts by setting the numerator equal to zero.

  3. Find the vertical asymptote by setting the denominator equal to zero.

  4. Determine the slant or horizontal asymptote by comparing degrees.

Try solving on your own before revealing the answer!

Q15. Graph:

Background

Topic: Graphing Rational Functions

This question involves factoring, finding intercepts, asymptotes, and sketching the graph of a rational function.

Key Terms and Formulas:

  • Same as above: factor, find intercepts, asymptotes, and analyze behavior.

Step-by-Step Guidance

  1. Factor the numerator if possible.

  2. Find the y-intercept by evaluating .

  3. Find the x-intercepts by setting the numerator equal to zero.

  4. Find the vertical asymptote by setting the denominator equal to zero.

  5. Determine the slant or horizontal asymptote by comparing degrees.

Try solving on your own before revealing the answer!

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