Q1. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your understanding of rational functions, including how to find holes (removable discontinuities), intercepts, horizontal asymptotes, and domain restrictions. These are key concepts in precalculus and college algebra.

Key Terms and Formulas:

• Hole: Occurs where a factor cancels in numerator and denominator.

• Intercepts: x-intercepts are where ; y-intercept is .

• Horizontal Asymptote: Determined by degrees of numerator and denominator.

• Domain: All real numbers except where denominator is zero.

Step-by-Step Guidance

1. Factor both the numerator and denominator completely. Look for common factors that may indicate holes.

2. Identify values of that make the denominator zero. These are potential domain restrictions and vertical asymptotes, unless canceled by a numerator factor (which would create a hole).

3. Find the x-intercepts by setting the numerator equal to zero and solving for (excluding any values that are holes).

4. Find the y-intercept by evaluating , if is in the domain.

5. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, asymptote is ; if greater, there is no horizontal asymptote.

Try solving on your own before revealing the answer!

Q2. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question focuses on analyzing a rational function with a constant numerator and quadratic denominator. You'll need to find domain restrictions, intercepts, and asymptotes.

Key Terms and Formulas:

• Vertical Asymptote: Where denominator is zero and not canceled.

• Horizontal Asymptote: If degree of numerator is less than denominator, .

• Intercepts: x-intercepts occur when numerator is zero; y-intercept is .

Step-by-Step Guidance

1. Factor the denominator to find values where the function is undefined.

2. Set the numerator equal to zero to find x-intercepts. Since the numerator is 1, consider if there are any x-intercepts.

3. Find the y-intercept by evaluating , if is in the domain.

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

Try solving on your own before revealing the answer!

Q3. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your ability to factor polynomials, identify holes, and analyze intercepts and asymptotes for a rational function.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• Domain: Exclude values where denominator is zero.

Step-by-Step Guidance

1. Factor both numerator and denominator completely. Look for common factors.

2. Identify values of that make the denominator zero. Check if any are canceled by numerator factors (holes).

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!

Q4. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question involves analyzing a rational function with a cubic denominator and linear numerator. You'll need to factor, find holes, and analyze intercepts and asymptotes.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Vertical Asymptote: Where denominator is zero and not canceled.

• Horizontal Asymptote: Degree of numerator less than denominator means .

Step-by-Step Guidance

1. Factor the denominator completely. Look for common factors with numerator.

2. Identify values of that make denominator zero. If $x$ is also a factor in numerator, it creates a hole.

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees of numerator and denominator.

Try solving on your own before revealing the answer!

Q5. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your skills in factoring, identifying holes, and analyzing intercepts and asymptotes for a rational function.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Horizontal Asymptote: Ratio of leading coefficients if degrees are equal.

• Domain: Exclude values where denominator is zero.

Step-by-Step Guidance

1. Factor numerator and denominator completely. Look for common factors.

2. Identify values of that make denominator zero. Check if any are canceled by numerator factors (holes).

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!

Q6. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question involves a rational function with linear numerator and denominator. You'll analyze intercepts, asymptotes, and domain.

Key Terms and Formulas:

• Vertical Asymptote: Where denominator is zero.

• Horizontal Asymptote: Ratio of leading coefficients if degrees are equal.

• Intercepts: x-intercept is where numerator is zero; y-intercept is .

Step-by-Step Guidance

1. Set denominator to find domain restriction and vertical asymptote.

2. Set numerator to find x-intercept.

3. Find y-intercept by evaluating .

4. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!

Q7. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your ability to factor, find holes, and analyze intercepts and asymptotes for a rational function.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• Domain: Exclude values where denominator is zero.

Step-by-Step Guidance

1. Factor numerator and denominator completely. Look for common factors.

2. Identify values of that make denominator zero. Check if any are canceled by numerator factors (holes).

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!

Q8. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your ability to factor, find holes, and analyze intercepts and asymptotes for a rational function.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• Domain: Exclude values where denominator is zero.

Step-by-Step Guidance

1. Factor numerator and denominator completely. Look for common factors.

2. Identify values of that make denominator zero. Check if any are canceled by numerator factors (holes).

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!

Q9. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question involves a rational function with a constant numerator and quadratic denominator. You'll analyze domain, intercepts, and asymptotes.

Key Terms and Formulas:

• Vertical Asymptote: Where denominator is zero.

• Horizontal Asymptote: Degree of numerator less than denominator means .

• Intercepts: x-intercept is where numerator is zero; y-intercept is .

Step-by-Step Guidance

1. Factor denominator to find domain restrictions and vertical asymptotes.

2. Set numerator equal to zero to find x-intercepts. Since numerator is constant, consider if there are any x-intercepts.

3. Find y-intercept by evaluating .

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

Try solving on your own before revealing the answer!

Q10. For the function , identify the holes, intercepts, horizontal asymptote, and domain. Then sketch the graph.

Background

Topic: Rational Functions

This question tests your ability to factor, find holes, and analyze intercepts and asymptotes for a rational function.

Key Terms and Formulas:

• Hole: Occurs when a factor cancels in numerator and denominator.

• Horizontal Asymptote: Degree of numerator less than denominator means .

• Domain: Exclude values where denominator is zero.

Step-by-Step Guidance

1. Factor numerator and denominator completely. Look for common factors.

2. Identify values of that make denominator zero. Check if any are canceled by numerator factors (holes).

3. Find x-intercepts by setting numerator equal to zero and solving for (excluding holes).

4. Find y-intercept by evaluating , if is in the domain.

5. Determine horizontal asymptote by comparing degrees and leading coefficients.

Try solving on your own before revealing the answer!