Most exam questions combine two transformations. Sometimes the order does not matter, and sometimes it changes the answer. Cambridge expects you to say which, and to say the order.
Do Now: the six moves, from memory5 min
Reading the notation, and the six moves on a graph7 min
Two moves: different directions, then the same direction14 min
Reading an equation, both routes, and your turn at the board14 min
The Order Lab, then a real 5-mark question from June 202415 min
Spot the mistake, practice, exit ticketflex
Do Now · 5 min · from last class
Say what each of these does to the graph of \(y = f(x)\). Tap to check.
\(4f(x)\) → stretch in the \(y\)-direction, factor 4 · \(f(3x)\) → stretch in the \(x\)-direction, factor \(\tfrac{1}{3}\)
\(-f(x)\) → reflection in the \(x\)-axis · \(f(-x)\) → reflection in the \(y\)-axis
Direct instruction · 1 of 10 · reading the notation
Three words, one column vector, and how to read the graphs.
The column vector
\(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) means right 3, down 2
top number is across (the \(x\)-direction): positive is right, negative is left
bottom number is up (the \(y\)-direction): positive is up, negative is down
so these three read: \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\) up 3 · \(\begin{pmatrix} -2 \\ 0 \end{pmatrix}\) left 2 · \(\begin{pmatrix} 4 \\ -1 \end{pmatrix}\) right 4, down 1
It is not a fraction and there is no line between the numbers. Across first, then up.
\(y = x^2\) translated by \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\)
The lowest point moves across 3, then down 2, from \((0, 0)\) to \((3, -2)\). Every other point moves the same way.
Translation
Slides the whole graph. No turning, no resizing. Described by a column vector.
Stretch
Pulls the graph away from an axis. The scale factor says how many times as far, and you must say which direction.
Reflection
Mirrors the graph in a line. You must name the line, the axis that stays fixed.
Dashed gray: before the move Solid cyan: after the move
Check · the six moves on a graph
The six moves, before and after, on one graph.
Dashed gray: before the move Solid cyan: after the move
Outside the bracket acts on \(y\). Hollow dots start, filled dots land.
Translation
\(f(x) + 3\)
Up 3, by \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\).
Stretch
\(4f(x)\)
\(y\)-direction, factor 4. Root stays put.
Reflection
\(-f(x)\)
In the \(x\)-axis. Root stays put.
Inside the bracket acts on \(x\), backwards.
Translation
\(f(x - 2)\)
Right 2, by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\).
Direct instruction · 2 of 10 · different directions
Different directions: the order does not matter.
\(y = x^2\), stretch in the \(y\)-direction, factor 2, and translate by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). Predict first: does the order change the final graph?
Dashed gray: before the move Solid cyan: after the move
Stretch first, then translate
Move 1 · Stretch
\(x^2 \to 2x^2\)
\(y\)-direction, factor 2: \((2, 4) \to (2, 8)\)
Move 2 · Translate
\(2x^2 \to 2(x - 3)^2\)
By \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\), right 3: \((2, 8) \to (5, 8)\)
Translate first, then stretch
Move 1 · Translate
\(x^2 \to (x - 3)^2\)
By \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\), right 3: \((2, 4) \to (5, 4)\)
Move 2 · Stretch
\((x - 3)^2 \to 2(x - 3)^2\)
\(y\)-direction, factor 2: \((5, 4) \to (5, 8)\)
Same final graph
\(y = 2(x - 3)^2\)
Both orders end at \((5, 8)\). One move acts on \(y\), the other on \(x\), so they never get in each other's way.
Direct instruction · 3 of 10 · same direction
Same direction: the order changes the graph.
\(y = x^2\): stretch in the \(y\)-direction, factor 2, and translate by \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\). Does the order matter now?
Dashed gray: before the move Solid cyan: after the move Dotted amber: other order
Stretch first, then translate up 3
Move 1 · Stretch
\(x^2 \to 2x^2\)
\(y\)-direction, factor 2: \((1, 1) \to (1, 2)\)
Move 2 · Translate
\(2x^2 \to 2x^2 + 3\)
by \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\), up 3: \((1, 2) \to (1, 5)\)
Translate up 3 first, then stretch
Move 1 · Translate
\(x^2 \to x^2 + 3\)
by \(\begin{pmatrix} 0 \\ 3 \end{pmatrix}\), up 3: \((1, 1) \to (1, 4)\)
Move 2 · Stretch
\(x^2 + 3 \to 2(x^2 + 3) = 2x^2 + 6\)
\(y\)-direction, factor 2: \((1, 4) \to (1, 8)\)
Two different final graphs
\(2x^2 + 3\) and \(2x^2 + 6\)
Cyan: stretch first, crosses the \(y\)-axis at 3. Amber: translate first, crosses at 6. A stretch multiplies everything already there, including an earlier shift.
Direct instruction · 4 of 10 · same direction, again
A reflection and a translation up also depend on order.
\(y = \sqrt{x}\): reflect in the \(x\)-axis, and translate by \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\). Follow \((4, 2)\).
Dashed gray: before the move Solid cyan: after the move Dotted amber: other order
Reflect first, then translate up 5
Move 1 · Reflect
\(\sqrt{x} \to -\sqrt{x}\)
in the \(x\)-axis: \((4, 2) \to (4, -2)\)
Move 2 · Translate
\(-\sqrt{x} \to 5 - \sqrt{x}\)
by \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\), up 5: \((4, -2) \to (4, 3)\)
Translate up 5 first, then reflect
Move 1 · Translate
\(\sqrt{x} \to \sqrt{x} + 5\)
by \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\), up 5: \((4, 2) \to (4, 7)\)
Cyan: reflect first, ends above the axis. Amber: translate first, ends below it. The flip reverses the earlier shift.
Direct instruction · 5 of 10 · reading an equation
An equation is a recipe. Read it in the order you would work it out.
\(y = 3(x + 2)^2 - 5\) is built from \(y = x^2\). What did we do to \(x^2\), and in what order?
Dashed gray: before the move Solid cyan: after the move
Start
\(y = x^2\)
Lowest point at \((0, 0)\).
Step 1 · Stretch
\(x^2 \to 3x^2\)
\(y\)-direction, factor 3. The lowest point stays at \((0, 0)\).
Step 2 · Translate
\(3x^2 \to 3(x + 2)^2 - 5\)
By \(\begin{pmatrix} -2 \\ -5 \end{pmatrix}\). The lowest point goes to \((-2, -5)\).
How to read it
The 3 multiplies the bracket, so it is the stretch: \(y\)-direction, factor 3. The \(+2\) is inside the bracket, and \(x + 2 = x - (-2)\), so it moves the graph left 2. The \(-5\) is added last, so it moves the graph down 5.
A stretch in the \(y\)-direction, factor 3, followed by a translation by \(\begin{pmatrix} -2 \\ -5 \end{pmatrix}\).
Direct instruction · 6 of 10 · the other route
Cambridge accepts either order, as long as you state the order and your numbers match it.
The same graph, \(y = 3(x + 2)^2 - 5\), built the other way: translate first, then stretch.
Dashed gray: before the move Solid cyan: after the move
Step 1 · Translate
\(x^2 \to (x + 2)^2 - \tfrac{5}{3}\)
By \(\begin{pmatrix} -2 \\ -\frac{5}{3} \end{pmatrix}\). The lowest point goes to \((-2, -\tfrac{5}{3})\).
Step 2 · Stretch
\((x + 2)^2 - \tfrac{5}{3} \to 3(x + 2)^2 - 5\)
\(y\)-direction, factor 3. The \(-\tfrac{5}{3}\) is tripled to \(-5\).
Same final graph
\(y = 3(x + 2)^2 - 5\)
Same graph as the last slide, different recipe.
Why \(-\tfrac{5}{3}\) and not \(-5\)
The stretch comes second, so it triples whatever shift came first. We need the shift to end at \(-5\). So we start with \(-5 \div 3 = -\tfrac{5}{3}\).
What you can write
Route 1: a stretch in the \(y\)-direction, factor 3, then a translation by \(\begin{pmatrix} -2 \\ -5 \end{pmatrix}\). Route 2: a translation by \(\begin{pmatrix} -2 \\ -\frac{5}{3} \end{pmatrix}\), then a stretch in the \(y\)-direction, factor 3. The June 2024 mark scheme for the question on slide 12 lists both routes.
At the board · 6 min
Your turn. Describe it fully, in order.
\(y = \dfrac{1}{x}\) becomes \(y = \dfrac{2}{x - 1} + 3\). Describe fully a sequence of transformations, making clear the order. [4]
Find the multiplier, read the bracket, read the number added last. The dotted lines are the asymptotes: where did each one move to?
Asymptotes \(x = 0\) and \(y = 0\) move to \(x = 1\) and \(y = 3\). The point \((1, 1)\) goes to \((2, 5)\).
Tap to check once you have written the sequence.
the 2 multiplies, so it is the stretch: \(\dfrac{1}{x} \to \dfrac{2}{x}\) · \(x - 1\) is in the bracket, so right 1; the 3 is added last, so up 3: \(\dfrac{2}{x} \to \dfrac{2}{x - 1} + 3\)
A stretch in the \(y\)-direction, factor 2, followed by a translation by \(\begin{pmatrix} 1 \\ 3 \end{pmatrix}\).
other route: a translation by \(\begin{pmatrix} 1 \\ \frac{3}{2} \end{pmatrix}\) first, then a stretch in the \(y\)-direction, factor 2
Direct instruction · 7 of 10 · the Lab
Predict where the dots land. Then try \(q = 0\) or \(a = 1\).
Dashed gray: before the move Solid cyan: after the move Dotted amber: other order Hollow dots: \((1, 0)\), \((0, 1.5)\)
Direct instruction · 8 of 10 · June 2024 · Paper 12 · Q2
\(y = x^2\) becomes \(y = 4(x - 3)^2 + 8\). Describe fully a sequence of transformations, making clear the order. [5]
1 · read the target, multiply before add: \(y = 4\cdot(x - 3)^2 + 8\)
2 · stretch first, the 4: \(x^2 \to 4x^2\)
3 · then translate by \(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\): \(4x^2 \to 4(x - 3)^2 + 8\)
A stretch in the \(y\)-direction, factor 4, followed by a translation by \(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\).
Gray dashed: the start. Amber: after the stretch. Cyan: the final curve.
The other valid route: translate by \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\) first, giving \((x - 3)^2 + 2\). Then stretch in the \(y\)-direction, factor 4, giving \(4(x - 3)^2 + 8\). The shift is 2, not 8, because the stretch afterwards multiplies it by 4.
Direct instruction · 9 of 10 · where the 5 marks come from
Five separate facts, and the order is one of them.
B1 a stretch, factor 4, in the \(y\)-direction1 mark
B2 a translation with 3 in the \(x\)-direction and 8 in the \(y\)-direction (B1 if one of those three facts is missing)2 marks
M1 two translations, one in each direction, and one stretch, with nothing extra1 mark
A1 the order is correct and clear: the stretch comes before the vertical translation1 mark
What Cambridge says about answers like this
The June 2024 report gave full marks to answers as short as one sentence, and stresses correct terminology and that "the required order is clear". Two sentences is plenty. You do not need a diagram.
Spot the mistake · 4 min
Target: \(y = x^2 \to y = 4(x - 3)^2 + 8\). Three answers. What is wrong with each? Tap to check.
A. "A translation by \(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\), then a stretch in the \(y\)-direction, factor 4."
Sign error. \((x - 3)\) means right 3, so the vector has +3.
Practice · together · 6 min
Describe fully, in order. Tap to check.
1. \(y = x^2\) becomes \(y = 3(x + 2)^2 - 5\).
multiply, then add: \(3 \cdot (x + 2)^2 - 5\)
A stretch in the \(y\)-direction, factor 3, then a translation by \(\begin{pmatrix} -2 \\ -5 \end{pmatrix}\).
other route: translate by \(\begin{pmatrix} -2 \\ -\frac{5}{3} \end{pmatrix}\) first, then stretch, \(y\)-direction, factor 3
2. \(y = x^2\) becomes \(y = 5 - x^2\).
rewrite it so the flip is visible: \(y = -x^2 + 5\)
A reflection in the \(x\)-axis, then a translation by \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\).
other route: translate by \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\) first, then reflect in the \(x\)-axis
Practice · on your own · 8 min
Now go from the description to the equation.
3. \(y = x^3\) is stretched in the \(y\)-direction, factor 2, then translated by \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\). Write the new equation. Would the other order change it?
\(y = 2(x + 1)^3\). No, the moves are in different directions.
4. \(y = x^3\) is translated by \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\), then stretched in the \(y\)-direction, factor 2. Write the new equation. What if the order were reversed?
5. \(y = x^2 - 4x\) passes through \((4, 0)\). It is stretched in the \(y\)-direction, factor 3, and translated by \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\). Where does \((4, 0)\) go with the stretch first? With the translation first?
\((4, -2)\) or \((4, -6)\), depending on the order.
Direct instruction · 10 of 10 · recap
Three things to carry out of here.
Order matters only when both moves act on the same variable. A \(y\)-stretch with an \(x\)-translation: no. A \(y\)-stretch with a \(y\)-translation, or a flip with a \(y\)-translation: yes, because the second move acts on the shift made by the first.
Read \(y = af(x - p) + q\) as stretch first, then translate by \(\begin{pmatrix} p \\ q \end{pmatrix}\). Built the other way round, the vertical shift becomes \(\dfrac{q}{a}\).
A full description has four parts. The type, the size, the direction or axis, and the order. Write the word "then".
Exit ticket · 5 min
Schoology, Paper 1 → 1.2 Functions.
Do the working on paper first. Check the due time on the assessment page.
Press Submit before you close the tab. An attempt left open scores nothing.
Homework · and where to practice more
Before next class.
Redo any practice question you did not get to. For each one, write the order down and test the other order to see whether it changes the answer.
Take any two moves from last class and describe them both ways: first one first, then the other first. Say which pairs give the same graph.