Analysis only. The guide says candidates are expected to change from one form to another, so the rearranging is the content rather than a preliminary to it.
Step through all four views without comment, then go back to the last one. The claimed vertex at (−3, −8) sits with the curve 72 above it, and the dashed line between them is drawn to scale. Nobody argues with that picture.
The axis of symmetry is drawn on every view, which is deliberate: it is the one feature all three forms agree about, and it is the check that refuses the sign error. Point at it on view 2, where the roots 1 and 5 are marked, and say that their average is the 3.
Worth one minute: ask which form you would want if a question asked for the minimum value, and which if it asked where the curve crosses the x-axis. Choosing the form before rearranging is the habit, and it is what stops a student completing the square on a question that wanted the roots.
| Question | Answer |
|---|---|
| 1. The y-intercept | 10, which is the c. |
| 2. The axis of symmetry | x = 3, from (1 + 5)/2 and from −b/(2a) = 12/4. |
| 3. The vertex's y-coordinate | f(3) = −8, which is the k. |
| 4. f(−3) | 18 + 36 + 10 = 64. |
| 5. Why the vertex is at x = +h | B. The squared bracket is least when it is zero, and x − h = 0 at x = h. |
Question 4 exists to make the sign error expensive rather than abstract. A student who has computed 64 themselves cannot then claim the curve passes through (−3, −8).
1 markThe correct form reached, or the feature identified.
1 markThe algebra, with the a taken out before completing the square.
1 markThe coordinates as a pair, with the signs right.
A vertex given as a single number loses the third mark. It is a point. Insist on brackets and a comma from the first example, because the habit is cheap now and the mark is not.
| They wrote | What happened |
|---|---|
| −3 on question 2 | The sign of h, read backwards. The minus is already printed in a(x − h)². Two checks refuse it, and both are one line. |
| 6 | Halved the −12 without taking the 2 out first. The commonest completing-the-square error, and it also puts the axis outside the roots, which is visible. |
| 1.5 | 12/8, using 2a as twice 2a. Arithmetic rather than method. |
| −4 on question 3 | The k before the 2 was multiplied back in: 2[(x − 3)² − 4]. They have done the hard part and stopped one step early. |
| 8 on question 3 | The sign of the k. Only the h has a minus in front of it in the formula; the k keeps its own. |
| −8 on question 4 | Read −12 × (−3) as −36. Two negatives give +36, so the terms are 18 + 36 + 10. |
| 28 | Took (−3)² as −9. Worth fixing here rather than in Topic 5. |
| 54 | 18 + 36 with the constant dropped. |
| 2 on question 1 | Gave the a. Ask what f(0) is. |
"Which form should I use?" Whichever has the answer written in it. Roots means factorised, vertex or minimum means completed square, y-intercept means general. If a question asks for two of those, you need two forms, and converting is quicker than it looks once the a is out.
"What if it will not factorise?" Then there are no real roots and the factorised form does not exist over the reals, but the vertex form always does. Do 2x² − 12x + 20 on the board: no roots, vertex (3, 2). That is also the cleanest motivation for the discriminant at 2.7, so it is worth spending the two minutes.
"Why take the a out first?" Because completing the square only works on x² + bx. Leave the 2 in and students halve −12 instead of −6. Show both on the board once and let the wrong one fail against the roots.
"Is the vertex always the minimum?" Only when a is positive. With a negative a it is the maximum, and the form is identical. Worth asking rather than telling: what does a < 0 do to the curve?
| Demonstrate | Enter all three forms as three separate functions and draw them. One curve appears. That is the fastest way to establish that rearranging has not changed anything, and it is worth doing before the algebra rather than after, so the class knows what they are aiming at. |
|---|---|
| Where they stick | The Nspire's Minimum wants bounds either side of the dip and students put both on the same side, which returns nothing. On the Casio, G-SOLVE is SHIFT F5 and ROOT needs the right arrow to reach the second root, so a class that stops at the first reports one root and a wrong axis. |
| The check | Two routes to the axis of symmetry: the average of the roots, and −b/(2a). They agree or something is wrong, and they are independent, so agreeing means something. The second works even when the first is unavailable. |
Analysis Paper 1 has no calculator and completing the square is Paper 1 work. Graph the three forms to establish the idea, then put the machines away for the rearranging.
| Step | What |
|---|---|
| 1 | Three forms of the same function on the board. Graph all three; one curve. |
| 2 | What each form hands you, as three sentences. |
| 3 | Ask for the vertex of 2(x − 3)² − 8. Collect the answers. |
| 4 | Figure, final view. 72 apart. |
| 5 | The reason: the bracket is zero at x = 3. Four words, no convention. |
| 6 | Both checks on the axis, done out loud. |
| 7 | Completing the square with the a taken out, and once without, to show the failure. |
| 8 | 2x² − 12x + 20: no roots, still a vertex. Sets up 2.7. |
Do not say "the vertex form is a(x − h)² + k so the vertex is (h, k), remember to flip the sign". There is nothing to flip, and telling them to flip something is what produces −3. The minus is in the formula so that h reads off directly.
Do not say "completing the square is just a trick for finding the vertex". It is the proof of the quadratic formula at 2.7, and it is how 2(x − 3)² + 2 shows at a glance that a curve has no roots. Calling it a trick invites students to skip it when a graph is available, and then Paper 1 arrives.