The shape of the lesson
By the end of the lesson, learners can
- Straight lines and parabolas: In the lab's sketching tab, change the coefficients and watch the key features move: the intercepts, the line of symmetry and the turning point.
- Cubic curves: Extended.
- Reciprocals, exponentials and asymptotes: Extended.
- Shape: decide from the highest power and its sign (U or ∩; rising or falling cubic; two-branch reciprocal; exponential).
- \(y\)-intercept: put \(x = 0\).
- Roots: put \(y = 0\) and solve, usually by factorising.
- Quadratic turning point (Extended): complete the square, \(y = (x + p)^2 + q\) turns at \((-p, q)\).
- Asymptotes (Extended): dashed lines, labelled; the curve gets close but never touches.
- Label every key point with its coordinates.
One picture every idea returns to
Sketch = correct shape + labelled key points. Set \(x = 0\) for the \(y\)-intercept and \(y = 0\) for the roots; for a quadratic the line of symmetry is halfway between the roots, and completing the square gives the turning point.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, The shape, the key points.. A sketch is not a plot. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Straight lines and parabolas | In the lab's sketching tab, change the coefficients and watch the key features move: the intercepts, the line of symmetry and the turning point. Project tab 2 of the lab (Graphs of functions lab, sketching) and let a student drive it while the class predicts. Model the worked example Sketching a straight line on the board, one line at a time. Your turn: Sketch \(y = -x^2 + 2x + 8\). Show the roots, the \(y\)-intercept and the turning point. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Cubic curves | Extended. Model the worked example Roots of a cubic (Extended) on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Reciprocals, exponentials and asymptotes | Extended. Model the worked example A reciprocal graph (Extended) on the board, one line at a time. | Frayer Model |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 18 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Reciprocals, exponentials and asymptotes, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Frayer Model
Pairs fill a four-box grid for the key term: definition, characteristics, examples and non-examples. The non-examples box is where the misconceptions surface, so give it the most time.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| In \((x - 3)^2 - 4\) the turning point is \((3, -4)\), not \((-3, -4)\). | The sign inside the bracket flips; the number outside does not. |
| Do not forget the root \(x = 0\) when you take out a factor of \(x\). \(x^3 + x^2 - 6x = 0\) has THREE solutions, not two. | Return to Cubic curves: ask a pair to state the correct version and show it on the lab before moving on. |
| \(y = 2^x\) does NOT pass through the origin: \(2^0 = 1\). | And an exponential curve never crosses its asymptote, so your sketch must not touch the dashed line. |
Sketching curves: roots, intercepts and turning points: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
Sketch the line \(3x + 4y = 24\), labelling where it crosses each axis.
Sketch \(y = x^2 - 2x - 8\). Label the roots, the \(y\)-intercept and the line of symmetry.
(Extended) (a) Write \(x^2 - 10x + 18\) in the form \((x + p)^2 + q\).
(b) Hence write down the turning point of \(y = x^2 - 10x + 18\), and state whether it is a maximum or a minimum.
(Extended) Sketch \(y = 4x - x^3\), showing where the curve crosses the axes.
(Extended) Sketch \(y = \dfrac{6}{x} - 3\). Give the equations of the asymptotes and the coordinates of any point where the curve crosses an axis.
(Extended) A cup of coffee cools so that its temperature, \(T\) °C, after \(t\) minutes is \(T = 70 \times 0.8^{t} + 20\).
(a) Find the temperature when \(t = 0\) and when \(t = 3\).
(b) State the equation of the asymptote of the graph and explain what it means for the coffee.
(c) Sketch the graph for \(t \ge 0\).
Answer key and marking notes
- \(x = 0\): \(4y = 24\), \(y = 6\), so \((0, 6)\).
- \(y = 0\): \(3x = 24\), \(x = 8\), so \((8, 0)\). Join with a straight line.
- \(x^2 - 2x - 8 = (x - 4)(x + 2)\), so roots \(x = 4\) and \(x = -2\).
- \(y\)-intercept: \(-8\).
- Line of symmetry halfway between the roots: \(x = 1\). U-shaped curve.
- (a) \(x^2 - 10x + 18 = (x - 5)^2 - 25 + 18 = (x - 5)^2 - 7\).
- (b) Turning point \((5, -7)\).
- The \(x^2\) term is positive (U shape), so it is a minimum.
- \(4x - x^3 = x(4 - x^2) = x(2 - x)(2 + x)\).
- Roots: \(x = -2, 0, 2\). The \(y\)-intercept is 0.
- The \(x^3\) term is negative, so the curve comes down from top-left, turns, passes through the origin, turns again and falls to bottom-right.
- Asymptotes: \(x = 0\) and \(y = -3\).
- \(x\)-intercept: \(\dfrac{6}{x} = 3\), so \(x = 2\): the point \((2, 0)\).
- There is no \(y\)-intercept because \(x = 0\) is not allowed. Branches lie top-right and bottom-left of the point where the asymptotes cross.
- (a) \(t = 0\): \(70 \times 1 + 20 = 90\) °C.
- \(t = 3\): \(70 \times 0.512 + 20 = 55.84 \approx 55.8\) °C.
- (b) As \(t\) increases, \(0.8^t \to 0\), so \(T \to 20\): asymptote \(T = 20\). The coffee cools towards room temperature, 20 °C, but never drops below it.
- (c) A decreasing curve starting at \((0, 90)\), falling steeply at first, then levelling off above the dashed line \(T = 20\).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Where does the line \(3x - 4y = 12\) cross the axes?
- A\((4, 0)\) and \((0, -3)\)
- B\((4, 0)\) and \((0, 3)\)
- C\((3, 0)\) and \((0, -4)\)
- D\((-4, 0)\) and \((0, 3)\)
The sketch shows a quadratic curve crossing the \(x\)-axis at \(-1\) and 3 and the \(y\)-axis at 3. Which could be its equation?
- A\(y = (x + 1)(3 - x)\)
- B\(y = (x + 1)(x - 3)\)
- C\(y = (1 - x)(x + 3)\)
- D\(y = (x - 1)(x + 3)\)
What is the equation of the line of symmetry of the graph of \(y = x^2 - 2x - 15\)?
- A\(x = 1\)
- B\(x = -1\)
- C\(x = 4\)
- D\(y = 1\)
By completing the square, find the turning point of \(y = 3 + 4x - x^2\) and its nature.
- AMaximum at \((2, 7)\)
- BMinimum at \((2, 7)\)
- CMaximum at \((-2, 7)\)
- DMaximum at \((2, -1)\)
What are the equations of the asymptotes of the graph of \(y = \dfrac{3}{x} - 2\)?
- A\(x = 0\) and \(y = -2\)
- B\(x = 0\) and \(y = 2\)
- C\(x = -2\) and \(y = 0\)
- D\(x = 3\) and \(y = -2\)
The sketch shows a cubic curve. It crosses the \(x\)-axis at \(-2\), 1 and 3 and the \(y\)-axis at \(-6\). Which is its equation?
- A\(y = -(x + 2)(x - 1)(x - 3)\)
- B\(y = (x + 2)(x - 1)(x - 3)\)
- C\(y = -(x - 2)(x + 1)(x + 3)\)
- D\(y = (x - 2)(x + 1)(x + 3)\)
Quiz key
\(x = 0\): \(-4y = 12\), so \(y = -3\), giving \((0, -3)\). \(y = 0\): \(3x = 12\), so \(x = 4\), giving \((4, 0)\). \((0, 3)\) loses the minus sign in \(-4y\). \((3, 0)\) and \((0, -4)\) swaps the numbers: you divide 12 by the coefficient, you do not read the coefficients off.
\((x + 1)(3 - x) = 0\) gives roots \(-1\) and 3; at \(x = 0\), \(y = 1 \times 3 = 3\); and it expands to \(-x^2 + 2x + 3\), a negative \(x^2\) term, so the curve is ∩-shaped, as in the sketch. \((x + 1)(x - 3)\) has the right roots but is U-shaped with \(y\)-intercept \(-3\). \((1 - x)(x + 3)\) has \(y\)-intercept 3 and the right shape, but its roots are 1 and \(-3\): the signs inside the brackets are flipped.
\(x^2 - 2x - 15 = (x - 5)(x + 3)\), so the roots are \(x = 5\) and \(x = -3\). Halfway: \(\dfrac{5 + (-3)}{2} = 1\), so the line is \(x = 1\). \(x = -1\) comes from roots taken as \(-5\) and 3 (signs flipped). \(x = 4\) is \(\dfrac{5 + 3}{2}\), ignoring the negative root. The line of symmetry of a parabola like this is vertical, so its equation is \(x = \dots\), never \(y = \dots\).
\(-(x^2 - 4x) + 3 = -[(x - 2)^2 - 4] + 3 = -(x - 2)^2 + 4 + 3 = 7 - (x - 2)^2\). The largest value is 7, when \(x = 2\), and the \(-x^2\) term makes the curve ∩-shaped, so \((2, 7)\) is a maximum. \((2, -1)\) comes from \(-(x - 2)^2 - 4 + 3\), forgetting that the minus outside the bracket changes \(-4\) into \(+4\). \((-2, 7)\) flips the sign of the \(x\)-coordinate.
You cannot divide by 0, so the curve never meets \(x = 0\) (the \(y\)-axis). As \(x\) gets very large, \(\dfrac{3}{x}\) gets close to 0, so \(y\) gets close to \(0 - 2 = -2\). The asymptotes are \(x = 0\) and \(y = -2\). \(y = 2\) loses the sign; \(x = -2\) puts the shift on the wrong axis: the \(-2\) is added to \(y\), so it moves the graph down, not sideways.
Roots \(-2\), 1, 3 give \((x + 2)(x - 1)(x - 3)\). The curve starts top-left and ends bottom-right, so the \(x^3\) term is negative and the whole product needs a minus sign. Check: at \(x = 0\), \(y = -(2)(-1)(-3) = -6\), as shown. \((x + 2)(x - 1)(x - 3)\) has the right roots but rises from bottom-left, and its \(y\)-intercept is \(+6\). The other two options have the signs in the brackets flipped, giving roots 2, \(-1\) and \(-3\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.