The shape of the lesson
By the end of the lesson, learners can
- Translations: A translation slides the whole graph without changing its shape.
- Reflections and stretches: Reflections flip the graph in an axis; stretches pull it away from an axis.
- Combining transformations: Two transformations in DIFFERENT directions (one on \(x\), one on \(y\)) can be done in either order.
- Translation \(\begin{pmatrix}p\\q\end{pmatrix}\): \(y = f(x - p) + q\).
- Reflection: \(-f(x)\) in the \(x\)-axis; \(f(-x)\) in the \(y\)-axis.
- Stretch: \(af(x)\) parallel to the \(y\)-axis, factor \(a\); \(f(ax)\) parallel to the \(x\)-axis, factor \(\frac{1}{a}\).
- Describe fully: name the transformation, the axis or direction, and the vector or factor.
- Combine one step at a time, rewriting the equation after each, to check the order.
One picture every idea returns to
Changes OUTSIDE the function act on \(y\) as you would expect: \(f(x) + a\) moves up \(a\), \(af(x)\) stretches by factor \(a\) parallel to the \(y\)-axis. Changes INSIDE act on \(x\) the opposite way: \(f(x + a)\) moves LEFT \(a\), \(f(ax)\) stretches by factor \(\dfrac{1}{a}\) parallel to the \(x\)-axis.
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, Same curve, new place.. Small changes to an equation move, flip or stretch its graph in predictable ways. 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: Translations | A translation slides the whole graph without changing its shape. Model the worked example Describe a translation on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Reflections and stretches | Reflections flip the graph in an axis; stretches pull it away from an axis. Model the worked example Following a point on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Combining transformations | Two transformations in DIFFERENT directions (one on \(x\), one on \(y\)) can be done in either order. Model the worked example Order on the outside on the board, one line at a time. Your turn: The point \((6, 4)\) lies on \(y = f(x)\). Find its image on \(y = 2f\left(\dfrac{x}{3}\right)\). | Numbered Heads Together |
| 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, 21 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 Combining transformations, 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.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
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 |
|---|---|
| \(y = f(x + 4)\) moves the graph 4 units LEFT, not right. | A point \((2, 5)\) on \(y = f(x)\) goes to \((-2, 5)\): you need \(x + 4 = 2\) to get the same output. |
| \(y = f(2x)\) is a stretch by factor \(\frac{1}{2}\), not 2: the graph is squeezed towards the \(y\)-axis. | And always say 'parallel to the \(x\)-axis' or 'parallel to the \(y\)-axis'; 'a stretch of factor 2' alone scores nothing. |
| Inside the bracket, translations and stretches interact: \(f(2x - 6)\) is NOT 'translate 6 right, stretch factor \(\frac{1}{2}\)' unless you do them in that order. | Test any sequence by building the equation. |
Transformations of graphs: worksheet
6 questions, 21 marks. Written in the style of the papers; no past paper question is reproduced.
Describe fully the single transformation that maps \(y = f(x)\) onto (a) \(y = f(x - 5)\) (b) \(y = \frac{1}{2}f(x)\).
The curve \(y = x^{2} - 3x\) is reflected in the \(y\)-axis and then stretched parallel to the \(y\)-axis with scale factor 2. Find the equation of the resulting curve.
The point P\((3, -2)\) lies on the curve \(y = f(x)\). Find the image of P on each curve.
(a) \(y = f(x) + 5\) (b) \(y = f(x - 1)\) (c) \(y = -2f(x)\) (d) \(y = f(3x)\)
\(f(x) = x^{2} - 6x + 11\).
(a) Express \(f(x)\) in the form \((x - a)^{2} + b\) and hence describe the transformation that maps \(y = x^{2}\) onto \(y = f(x)\).
(b) Find the coordinates of the minimum point of \(y = 2f(x) + 1\).
Show that the curve \(y = x^{2} + 4x + 1\) is a translation of the curve \(y = x^{2}\), and state the translation vector.
The curve \(y = f(x)\) has a maximum point at \((2, 5)\) and crosses the \(x\)-axis only at \((-1, 0)\) and \((4, 0)\).
For each curve, state the coordinates of its turning point and of the points where it crosses the \(x\)-axis, and say whether the turning point is a maximum or a minimum.
(a) \(y = f(x - 3)\) (b) \(y = 2f(x)\) (c) \(y = f(2x)\) (d) \(y = -f(x)\)
Answer key and marking notes
- (a) A translation by \(\begin{pmatrix}5\\0\end{pmatrix}\).
- (b) A stretch parallel to the \(y\)-axis with scale factor \(\frac{1}{2}\).
- Reflect in the \(y\)-axis: replace \(x\) by \(-x\): \(y = (-x)^{2} - 3(-x) = x^{2} + 3x\).
- Stretch parallel to the \(y\)-axis, factor 2: multiply by 2.
- \(y = 2x^{2} + 6x\).
- (a) Up 5: \((3,\ 3)\).
- (b) Right 1: \((4,\ -2)\).
- (c) Reflect in the \(x\)-axis and stretch by 2: \(y = -2 \times (-2) = 4\), so \((3,\ 4)\).
- (d) Stretch parallel to the \(x\)-axis, factor \(\frac{1}{3}\): \((1,\ -2)\).
- (a) \(f(x) = (x - 3)^{2} - 9 + 11 = (x - 3)^{2} + 2\).
- A translation by \(\begin{pmatrix}3\\2\end{pmatrix}\).
- (b) The minimum of \(y = f(x)\) is \((3, 2)\). The stretch doubles \(y\) and the translation adds 1: \(y = 2(2) + 1 = 5\).
- Minimum point \((3,\ 5)\) (check: \(2f(x) + 1 = 2(x - 3)^{2} + 5\)).
- Complete the square: \(x^{2} + 4x + 1 = (x + 2)^{2} - 4 + 1 = (x + 2)^{2} - 3\).
- This is \(f(x + 2) - 3\) with \(f(x) = x^{2}\): \(x\) replaced by \(x + 2\) and 3 subtracted.
- So it is a translation of \(y = x^{2}\) by \(\begin{pmatrix}-2\\-3\end{pmatrix}\).
- (a) Translation 3 right: maximum \((5, 5)\); crosses at \((2, 0)\) and \((7, 0)\).
- (b) \(y\)-values doubled: maximum \((2, 10)\); crosses at \((-1, 0)\) and \((4, 0)\) (zero doubled is still zero).
- (c) \(x\)-values halved: maximum \((1, 5)\); crosses at \(\left(-\frac{1}{2}, 0\right)\) and \((2, 0)\).
- (d) Reflection in the \(x\)-axis: the maximum becomes a MINIMUM at \((2, -5)\); crosses at \((-1, 0)\) and \((4, 0)\).
8 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.
Describe the transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}(x - 5)\).
- ATranslation by \(\begin{pmatrix}5\\0\end{pmatrix}\)
- BTranslation by \(\begin{pmatrix}-5\\0\end{pmatrix}\)
- CTranslation by \(\begin{pmatrix}0\\-5\end{pmatrix}\)
- DTranslation by \(\begin{pmatrix}0\\5\end{pmatrix}\)
Describe the transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}\left(\dfrac{x}{2}\right)\).
- AStretch parallel to the \(x\)-axis with factor 2
- BStretch parallel to the \(x\)-axis with factor \(\dfrac{1}{2}\)
- CStretch parallel to the \(y\)-axis with factor 2
- DStretch parallel to the \(y\)-axis with factor \(\dfrac{1}{2}\)
The point \((4,\ -3)\) lies on the curve \(y = \text{f}(x)\). Find the corresponding point on the curve \(y = \text{f}(-x)\).
- A\((-4,\ -3)\)
- B\((4,\ 3)\)
- C\((-4,\ 3)\)
- D\((4,\ -3)\)
The curve \(y = x^{2} - 3x\) is translated by \(\begin{pmatrix}-1\\2\end{pmatrix}\). Find the equation of the new curve in the form \(y = x^{2} + bx + c\).
- A\(y = x^{2} - x\)
- B\(y = x^{2} - 5x + 6\)
- C\(y = x^{2} - x - 4\)
- D\(y = x^{2} - x + 3\)
Which sequence of transformations maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = 2 - \text{f}(x)\)?
- AA reflection in the \(x\)-axis, followed by a translation by \(\begin{pmatrix}0\\2\end{pmatrix}\)
- BA translation by \(\begin{pmatrix}0\\2\end{pmatrix}\), followed by a reflection in the \(x\)-axis
- CA reflection in the \(y\)-axis, followed by a translation by \(\begin{pmatrix}0\\2\end{pmatrix}\)
- DA reflection in the \(x\)-axis, followed by a translation by \(\begin{pmatrix}0\\-2\end{pmatrix}\)
Which sequence of transformations maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}\left(\dfrac{1}{2}x + 1\right)\)?
- AA translation by \(\begin{pmatrix}-1\\0\end{pmatrix}\), followed by a stretch parallel to the \(x\)-axis with factor 2
- BA stretch parallel to the \(x\)-axis with factor 2, followed by a translation by \(\begin{pmatrix}-1\\0\end{pmatrix}\)
- CA translation by \(\begin{pmatrix}1\\0\end{pmatrix}\), followed by a stretch parallel to the \(x\)-axis with factor 2
- DA translation by \(\begin{pmatrix}-1\\0\end{pmatrix}\), followed by a stretch parallel to the \(x\)-axis with factor \(\dfrac{1}{2}\)
The point \((6,\ 4)\) lies on the curve \(y = \text{f}(x)\). Find the corresponding point on the curve \(y = 3\text{f}(x + 2) - 1\).
- A\((4,\ 11)\)
- B\((8,\ 11)\)
- C\((4,\ 9)\)
- D\((4,\ 12)\)
The curve \(y = x^{2} - 4x + 1\) is reflected in the \(y\)-axis and then stretched parallel to the \(y\)-axis with factor 2. Find the equation of the resulting curve.
- A\(y = 2x^{2} + 8x + 2\)
- B\(y = -2x^{2} - 8x - 2\)
- C\(y = 4x^{2} + 8x + 1\)
- D\(y = 2x^{2} - 8x + 2\)
Quiz key
\(\text{f}(x - 5)\) takes at \(x = a + 5\) the value that f took at \(x = a\), so every point moves 5 units to the RIGHT: translation by \(\begin{pmatrix}5\\0\end{pmatrix}\). The tempting \(\begin{pmatrix}-5\\0\end{pmatrix}\) reads the minus sign literally. The vertical translations belong to \(\text{f}(x) - 5\) and \(\text{f}(x) + 5\).
\(\text{f}\left(\dfrac{x}{2}\right) = \text{f}\left(\dfrac{1}{2}x\right)\), so \(a = \dfrac{1}{2}\) and the factor is \(\dfrac{1}{a} = 2\), parallel to the \(x\)-axis: the point \((p,\ q)\) moves to \((2p,\ q)\). The tempting factor \(\dfrac{1}{2}\) reads the number inside without inverting it. Stretches parallel to the \(y\)-axis come from multiplying OUTSIDE, \(a\,\text{f}(x)\).
\(y = \text{f}(-x)\) is a reflection in the \(y\)-axis: when \(x = -4\), \(\text{f}(-x) = \text{f}(4) = -3\). The point is \((-4,\ -3)\). The tempting \((4,\ 3)\) is the reflection in the \(x\)-axis, which belongs to \(y = -\text{f}(x)\). \((-4,\ 3)\) reflects in both axes, and \((4,\ -3)\) assumes nothing changes.
\(y = (x + 1)^{2} - 3(x + 1) + 2 = x^{2} + 2x + 1 - 3x - 3 + 2 = x^{2} - x\). The tempting \(x^{2} - 5x + 6\) replaces \(x\) by \(x - 1\), moving the curve right instead of left. \(x^{2} - x - 4\) subtracts the 2 instead of adding it, and \(x^{2} - x + 3\) replaces only the \(x^{2}\), leaving \(-3x\) unchanged.
Reflecting in the \(x\)-axis gives \(y = -\text{f}(x)\); then translating up 2 gives \(y = -\text{f}(x) + 2 = 2 - \text{f}(x)\). The tempting reverse order gives \(\text{f}(x) + 2\) and then \(-\big(\text{f}(x) + 2\big) = -\text{f}(x) - 2\), which is different. A reflection in the \(y\)-axis gives \(\text{f}(-x)\), and translating down 2 gives \(-\text{f}(x) - 2\).
Translating by \(\begin{pmatrix}-1\\0\end{pmatrix}\) gives \(\text{f}(x + 1)\); a stretch with factor 2 parallel to the \(x\)-axis then replaces \(x\) by \(\frac{1}{2}x\), giving \(\text{f}\left(\frac{1}{2}x + 1\right)\). The tempting stretch-first order gives \(\text{f}\left(\frac{1}{2}x\right)\), then \(\text{f}\left(\frac{1}{2}(x + 1)\right) = \text{f}\left(\frac{1}{2}x + \frac{1}{2}\right)\). Translating by \(+1\) gives \(\text{f}\left(\frac{1}{2}x - 1\right)\), and factor \(\frac{1}{2}\) gives \(\text{f}(2x + 1)\).
\(x + 2 = 6\) gives \(x = 4\), where \(\text{f}(x + 2) = \text{f}(6) = 4\). Then \(y = 3 \times 4 - 1 = 11\): the point is \((4,\ 11)\). The tempting \((8,\ 11)\) moves the point right instead of left. \((4,\ 9)\) subtracts 1 before multiplying by 3, and \((4,\ 12)\) forgets the \(-1\).
Reflecting: \(y = (-x)^{2} - 4(-x) + 1 = x^{2} + 4x + 1\). Stretching: \(y = 2(x^{2} + 4x + 1) = 2x^{2} + 8x + 2\). The tempting \(-2x^{2} - 8x - 2\) also reflects in the \(x\)-axis. \(4x^{2} + 8x + 1\) stretches parallel to the \(x\)-axis (replacing \(x\) by \(-2x\)) instead, and \(2x^{2} - 8x + 2\) forgets the reflection.
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.