• Quick retrieval from earlier learning (Rational functions and asymptotes; Find limits of rational functions as x → ∞). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Find the vertical asymptote of f(x) = (2x − 1)/(x + 5). Give the x-value. → -5Prompt: The function has no value where the denominator is zero.
2. Find the limit as x → ∞ of (−3x² + 8x + 3)/(7x² + 9x − 1). Give a fraction if needed. → -3/7Prompt: For huge x, only the highest powers matter.
3. What happens to (2x³ + 2)/(x² − 3) as x → ∞? (0 / 2 / No finite limit (it grows without bound)) → No finite limit (it grows without bound)Prompt: Compare the degree of the top with the degree of the bottom.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Find vertical and horizontal asymptotes. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: f(x) = (4x − 9)/(x − 1). Find the vertical asymptote x = ?
• Step 1: x = 1
• Answer: 1
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. f(x) = (−2x + 6)/(x − 5). Find the vertical asymptote x = ? → 5Prompt: A vertical asymptote is where the denominator is zero (and the top isn't).
2. What are the vertical asymptotes of f(x) = (3x² + 1)/(3x² − 27)? (y = 1 / There are none / x = 3 / x = 3 and x = −3) → x = 3 and x = −3Prompt: Set the denominator equal to zero.

• Students work on the student sheet. Circulate and check the first two answers of every student early.
• Students who finish: write a question of their own for a partner, with the answer.

The 6 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. What are the vertical asymptotes of f(x) = (2x² + 1)/(4x² − 64)? (x = 16 and x = −16 / x = 4 and x = −4 / x = 4 / y = 1/2) → x = 4 and x = −4Prompt: Set the denominator equal to zero.
2. f(x) = (4x + 5)/(x + 3). Find the horizontal asymptote y = ? → 4Prompt: For large x, f(x) behaves like the ratio of the x terms.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.