At a glance. One lesson of about 60 minutes. It builds on lessons 3 and 4. Students read asymptotes from y = k/(x − p) + q and from a single fraction, write one form as the other, and state domain and range.
Calculator: GDC optional. GDC optional: a graph confirms asymptotes and intercepts found by hand, but the reasoning is by hand.
What students will learn
Know
- y = k/(x − p) + q has asymptotes x = p and y = q.
- (ax + b)/(cx + d) has x = −d/c and y = a/c.
- Domain excludes the vertical asymptote, range excludes the horizontal one.
Understand
- A translation moves the asymptotes with the graph.
- The single-fraction form hides the shift, which division by the bottom reveals.
- The graph never meets its vertical asymptote and, for this family, never takes the horizontal asymptote's value.
Do
- Sketch a shifted reciprocal graph with its asymptotes.
- Write (ax + b)/(cx + d) as q + k/(x − p).
- Find intercepts and heights.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: asymptotes and intercepts (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record how each slider changes the asymptote lines.
- Criterion C. Communicating: giving the domain and range with correct notation.
- Criterion D. Applying mathematics in real-life contexts: the school trip cost (hook).
The one thing to do with the figure
Ask for the prediction first. Set p = 0 and q = 0, then move one slider at a time. Ask students which dashed line moves, and by how much.
Ask: “Which slider would you move to make the vertical asymptote x = −2?” Set p to −2 and read the new single fraction.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the school trip cost; collect where the asymptotes will be. |
| 12 min | Figure, key idea and warning card; sketch one shifted graph on mini whiteboards. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, solve x + 4 = 0 together. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | y = (x − 3)2; x = 0 and y = 0; (3x − 1) ÷ (x − 1). |
|---|---|
| Hook | The vertical asymptote is x = 1 (no paying students) and the horizontal asymptote is y = 3: the cost per student falls towards 3 thousand rupees but never reaches it. |
| Worked examples | x = −2 and y = −4, with intercepts 0.5 and 0.25; x = 3 and y = 2; domain x ≠ −3, range y ≠ 2. |
| Faded example | y = 3 − 1/(x + 1), with x = −1 and y = 3. |
| 1. Asymptotes of a single fraction | A. B has the wrong sign; C reads 3 from a constant; D inverts the ratio. |
| 2. Evaluating a shifted reciprocal | 8. 5/1 + 3. |
| 3. An x intercept | 2.5. 15 ÷ 6. |
| 4. A vertical asymptote | −5. 2x + 10 = 0. |
| 5. Finding a from a point | 6. a ÷ 2 = 3. |
| 6. Where a graph reaches a height | 3.5. 3x + 2 = 5x − 5. |
| 7. Using the range | −0.75. a = 8, then 8x + 6 = 0. |
| 8. Reading a translation | C. A moves right; B moves up; D subtracts 5 from the top only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used the wrong sign, read 3 from a constant, or inverted the ratio. | Ask: where is the bottom zero? What happens to y for large x? |
| 4 or 5 in question 2 | Used x + 2, or forgot the + 3. | Put the value in the bottom first. |
| −2.5 or −0.5 in question 3 | Sign slip, or gave the asymptote. | Ask: when is a fraction zero? |
| 5 or −10 in question 4 | Wrong sign, or only the constant. | Solve the bottom equals zero as an equation. |
| 3 or 12 in question 5 | Forgot the bottom, or used x + 1. | Substitute and keep the bottom. |
| −1.5 or 1 in question 6 | Sign slip in the expansion, or ignored the bottom. | Multiply both sides by the bottom, then expand. |
| −1.5 or 1.5 in question 7 | Took a as the asymptote, or gave the vertical asymptote. | Find a from a/2 = 4 first, then set the top to 0. |
| A, B or D in question 8 | Moved the wrong way across or up, or subtracted 5 from the top only. | Say which part of the rule moves left or right and which moves up or down. |
Questions worth asking
- Why does x + 2 in the bottom move the graph left? Listen for: the bottom is zero at x = −2.
- What does y approach as x gets large? Listen for: the ratio of the x terms.
- Can the graph cross its horizontal asymptote? Listen for: not for this family; the equation y = q has no solution.
- How can you find the intercepts quickly? Listen for: x = 0 for the y intercept; the top equal to 0 for the x intercept.
Differentiation
A way in for anyone stuck
- Give a table that links the rule to the two asymptotes for four shifted graphs.
- Provide axes with the asymptotes drawn and ask students to sketch the branches.
- Give the frame: “The bottom is zero when …. For large x, y gets close to ….”
Stretch
- Show that the graph of (ax + b)/(cx + d) is a straight line if ad = bc, and say what the line is.
- Describe the stretch and translation that map y = 1/x onto y = (2x + 5)/(x + 1).
Students learning in English as an additional language
“Translate” means move without turning or stretching. “Intercept” is where a graph crosses an axis. “Ratio” is one number divided by another. “Asymptote” is a line the graph gets close to.
| Word | Plain meaning | Example |
|---|---|---|
| translation | moving a graph without turning or stretching it | 4 right and 1 down |
| intercept | a point where a graph crosses an axis | where x = 0 |
| vertical asymptote | a vertical line the graph approaches | x = 0 for y = 1/x |
| horizontal asymptote | a horizontal line the graph approaches | y = 0 for y = 1/x |
| ratio of the x terms | the number in front of x on top divided by the number in front of x below | 8x over 4x is 2 |
Sentence starters for explaining aloud or in writing:
- The bottom is zero when …, so the vertical asymptote is ….
- As x gets large, y gets close to ….
- The graph is moved … across and … up.
Two things not to say
“x + 3 moves the graph right.” Say instead: it moves it 3 to the left, because the bottom is zero at x = −3.
“The asymptotes are always the axes.” Say instead: they move with the graph.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- AsymptotesState the asymptotes of y = 4/(x − 3) + 1.
x = 3 and y = 1. - EvaluateFind y at x = 5 for y = 6/(x − 2) − 1.
1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Single fractionState the asymptotes of y = (5x + 1)/(x − 4).
x = 4 and y = 5. - Earlier work (lesson 4)y = k/x passes through (−3, 4). Find k.
−12.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why y = (3x + 6)/(x + 2) is not a reciprocal graph.
It simplifies to y = 3, with a hole at x = −2. - Working backwardsy = (ax + 3)/(x + b) has asymptotes x = −2 and y = 4. Find a and b.
b = 2 and a = 4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
A GDC is optional and sits in a closed section; nothing depends on it.