Common content, so this runs for both courses.
Ask whether a graph can touch its asymptote, and wait. The answer will be a confident no, usually with the words "gets closer and closer but never reaches it". Write that sentence up, because it is half right and you are about to use the other half.
Then the first view: the curve sitting exactly on y = 0 at x = −0.5. Do the arithmetic in front of them rather than pointing at the picture. The numerator 2x + 1 is zero there, so the whole fraction is zero. Nothing approximate about it.
Then the second view, which is the part that stops it feeling like a trick: at x = 5 the value is 0.42 and at x = 50 it is 0.04. The asymptote was telling the truth all along. It was a claim about far away, and the crossing is not far away.
Views three and four do the vertical case and a horizontal one that genuinely is never reached, so the lesson is not "the rule is wrong" but "the rule was about something else".
| Question | Answer |
|---|---|
| 1. y of the vertex of x² − 4x + 1 | 4 − 8 + 1 = −3. |
| 2. Larger root | 2 + √3 = 3.73. |
| 3. Vertical asymptote of (2x+1)/(x−3) | x = 3. |
| 4. Maximum of (2x+1)/(x²+1) | 1.618, at x = 0.618. |
| 5. The accurate version of the claim | A. True for vertical, can be false for horizontal. |
1 markEach feature found, to the stated accuracy.
1 markEach feature correctly named, where a description is asked for.
1 markAn asymptote given as an equation.
That last one is the cheapest mark lost here. An asymptote is a line, so the answer is x = 3 and not 3. Mark it strictly in class once and it stops.
| They wrote | What happened |
|---|---|
| 2 on question 1 | Gave the x-coordinate of the vertex. |
| 1 | Gave the y-intercept, the value at x = 0 rather than at x = 2. |
| +3 | Lost a sign. Worth pointing out that it must be negative: the parabola has two real roots, so it dips below the axis. |
| 0.27 | On question 2, the smaller root. Both are right answers to a question that was not asked. |
| 3.74 | Rounded √3 to 1.74 before adding. The rounding habit worth stopping here, because it costs a mark and nothing else about the work is wrong. |
| 2 on question 3 | Gave the horizontal asymptote. They have both, and have paired them the wrong way round. |
| −0.5 on question 3 | Gave the x-intercept. Top zero and bottom zero do very different things. |
| 0.618 | On question 4, gave where the maximum is rather than what it is. |
| 1.5 | On question 4, read the value at x = 1 off a table. Close to the top and not at it. |
| 1.62 | Right value, wrong precision. The question asked for three decimal places. |
"So the rule is just wrong?" No: it is right about vertical asymptotes, where the function genuinely has no value, and it was never a statement about horizontal ones. Give them the test instead of the rule: try to solve f(x) = L. If it has a solution the curve meets the line there; if it gives you 1 = −6 it never does.
"Why does the figure draw the curve in two pieces?" Because it is two pieces. A single path through a vertical asymptote draws a near-vertical line that looks like part of the graph and is not. Worth showing on a calculator too, where most machines do exactly that and lie to the student.
"Is the golden ratio thing examinable?" No, and say so plainly. The maximum of (2x+1)/(x²+1) being (1 + √5)/2 is a coincidence of the numbers chosen. It is there because noticing things is worth encouraging, not because it is on a syllabus.
| Stage | What to do |
|---|---|
| Demonstrate | Find the maximum with the tool and write 1.618 up. Then read the same point off the screen by eye and write 1.6 beside it. Ask which one they would put in an exam asking for three significant figures. The machine knew; the picture did not. |
| Where they stick | Every tool here answers once per go, with bounds. A question worth two marks for "the zeros" needs the tool run twice, and students run it once and move on. Also brackets: entering (2x+1)/x²+1 without the outer bracket graphs something else entirely. |
| The check | Ask how many of each feature to expect before using any tool. A parabola has one vertex, up to two roots and one y-intercept. A student who expects two roots will not stop at one. |
On the Casio, G-SOLVE is SHIFT F5. Pressing F5 alone gets a different menu, and the students who find that out in an exam lose the time as well as the mark.
| Step | What |
|---|---|
| 1 | "Can a graph touch its asymptote?" Write their sentence up. |
| 2 | View one, with the arithmetic done on the board, not pointed at. |
| 3 | View two: it still tends to zero. The claim was about far away. |
| 4 | Views three and four: the vertical case, and a horizontal one it never meets. |
| 5 | The test, f(x) = L, replacing the rule. |
| 6 | The feature vocabulary on a parabola, where everything is findable by hand. |
| 7 | Zero, root and x-intercept as three words for one place. |
Do not say "it gets closer and closer but never touches". It is the sentence this lesson exists to replace, and if it comes from you it will outlast anything else you say today.
Do not let an asymptote be answered as a number. "3" is not a line. Insist on x = 3 from the first example, because it is a mark and it is free.