Transforming graphs Why does adding inside the bracket move the curve the opposite way?

The lesson, step by step

30 minutes in total. Each step says what changes for this length.

1. 1 · Think3 min

   Which way will it move?

   y = x² becomes y = (x − 2)².

   Left or right? By how much?

   And y = x² − 2?

   Predict both. Say why.

   Write here

   3 min 1 · Hook · students think

   Which way will it move?

   Students predict two moves before anyone explains.

   For this length: Vote on (x − 2)² only; skip the second prediction

   Say “y = x² becomes y = (x − 2)². Left or right, and by how much? And y = x² − 2? Write your predictions.”

   Ask

   • Where is (x − 2)² zero?
   • What is y = x² − 2 when x is 0?

   Watch for Predicting left 2 for (x − 2)²: keep it. The next step is about exactly that.

   For the teacher

   Give a minute of silent writing, then vote: left or right? Many will say left, reading the minus as a direction.

   Don't explain yet. The question is the point: one change goes the way it looks, the other the opposite way. Why?

   Take a vote with Ask the class for (x − 2)²: left or right. A split class makes the next step matter.

   Shorter: Vote on (x − 2)² only; skip the second prediction (−2 min).

2. 2 · Watch8 min

   Outside: y = f(x) + a

   y = x² + a

   Every y goes up a.

   The vector (0, a)

   As a column: 0 across, a up

   Predict, then move a.

   2 · Watch

   Inside: y = f(x + a)

   y = (x + a)²

   It is 0 at x = −a, not at 0.

   Each point moves from x to x − a:

   the vector (−a, 0)

   Inside works the opposite way.

   2 · Watch

   Reflections

   f(x) = x² − 2x

   −f(x) = −x² + 2x:

   reflected in the x-axis

   f(−x) = x² + 2x:

   reflected in the y-axis

   A minus outside flips y; inside, it flips x.

   2 · Watch

   Follow one point

   (2, 4) is on y = f(x)

   y = f(x) + 3: (2, 7)

   y = f(x + 3): (−1, 4)

   y = −f(x): (2, −4)

   y = f(−x): (−2, 4)

   Outside changes y. Inside changes x, the opposite way.

   8 min 2 · Explain · you explain

   Outside, inside, reflections

   Move a outside, then inside the bracket; then the reflections; then one point through each.

   For this length: Translations only; the reflections as a slide to read

   Say “Outside the bracket changes y, the way it looks. Inside changes x, the opposite way.”

   Ask

   • Before I move a to 3 outside: where will the lowest point be?
   • y = (x + 2)²: at which x is it 0? So which way did it move?

   Watch for Moving f(x + a) right: the curve reaches each height at a smaller x, so it is a to the left.

   Inside: where is it zero?

   y = (x + 2)² is 0 when x = −2

   y = x² is 0 when x = 0

   So it moved 2 left: vector (−2, 0)

   Reflect: f(x) = x² − 2x

   −f(x) = −x² + 2x: in the x-axis

   f(−x) = x² + 2x: in the y-axis

   For the teacher

   First slide: y = x² + a. Every y gets a added, so the curve moves up a: the vector (0, a). Move a; the reference curve y = x² stays.

   Second slide: y = (x + a)². It is 0 when x = −a, where y = x² was 0 at x = 0: each point moves from x to x − a, so the curve moves left a, the vector (−a, 0). This answers the hook's vote.

   Third slide: −f(x) changes the sign of every y, a reflection in the x-axis; f(−x) changes the sign of every x, a reflection in the y-axis. Then follow one point, (2, 4), through each change.

   Vote before each move (Ask the class): a to 3 outside; a to 3 inside.

   Shorter: Translations only; the reflections as a slide to read (−4 min).

3. 3 · Your turn5 min

   Worked, then yours

   I do: y = x² moved 3 right

   Inside, the opposite sign: y = (x − 3)²

   You do: y = x² moved 4 down

   Outside or inside?

   Write here

   3 · Your turn

   Two more pairs

   I do: (1, 5) on y = f(x) goes to (3, 5) on y = f(x − 2)

   You do: (1, 5) on y = f(x). Where on y = f(x) − 2?

   I do: y = x² reflected in the x-axis, then 4 up

   y = −x² + 4

   You do: y = x² moved 1 left, then 2 up

   Write here

   Its answers' slide comes with your copy.

   5 min 3 · Your turn · students write

   Worked, then yours

   Each worked example beside a twin that students solve themselves.

   For this length: Do the first two pairs only

   Say “Watch me do one. Then you do its twin, which is almost the same.”

   Ask

   • Outside or inside? So does x or y change?

   Watch for Changing both coordinates of a point when only one moves.

   An equation from a move

   y = x² moved 3 right

   Inside, the opposite sign: y = (x − 3)²

   Check: it is 0 at x = 3

   Then theirs

   y = x² moved 4 down

   Follow a point

   (1, 5) is on y = f(x)

   y = f(x − 2): inside, x changes the opposite way

   (1, 5) goes to (3, 5)

   Then theirs

   (1, 5) on y = f(x): where on y = f(x) − 2?

   Two moves

   y = x² reflected in the x-axis, then up 4

   y = −x² + 4

   It meets the x-axis at x = −2 and x = 2

   Then theirs

   y = x² moved 1 left, then 2 up

   For the teacher

   Work the first example aloud, writing each line. Then students solve its twin while you watch who changes the sign inside the bracket.

   Do the same for each pair. The second pair catches students who change both coordinates.

   Open the answers slide once most have finished their twins.

   Shorter: Do the first two pairs only (−3 min).

4. 4 · Vote and practise9 min

   Vote first

   y = x² moves to y = (x − 4)².

   How did it move?

   A: 4 to the right

   B: 4 to the left

   C: 4 up

   D: 4 down

   Choose on your own, then say why.

   4 · Vote and practise

   Practise: say what changes

   1. y = f(x) + 5: which way?

   2. (3, 2) on y = f(x): where on y = f(x) − 4?

   3. (3, 2) on y = f(x): where on y = f(x + 1)?

   4. y = x² moved 3 left: its equation?

   Stuck on 3? Inside: x changes, the opposite way.

   Write here

   4 · Vote and practise

   Reason it out

   5. Reflect y = x² in the y-axis: what happens?

   6. y = −f(x) reflects in which axis?

   7. x² − 6x + 11 = (x − 3)² + 2: which vector?

   Stretch: lowest point (1, −3). Where on y = f(x − 2) + 5?

   9 min 4 · Who's stuck? · students vote

   Vote, then practise

   One question shows who's stuck; then practice in levels, and a stretch.

   For this length: After the vote, questions 1–4 only

   Say “Choose an answer on your own. Nobody talks until everyone has chosen.”

   Ask

   • Where is the new curve 0? Where was the old one?

   Watch for Moving the curve the way the sign looks.

   Show your working: for each move, say whether x or y changes. The stretch is optional.

   The question that shows who's stuck

   1. y = x² is moved to y = (x − 4)². How did it move?
      1. 4 to the right
      2. 4 to the left
      3. 4 up
      4. 4 down

   • A9
   • B11
   • C3
   • D2

   Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.

   1. y = f(x) + 5 is y = f(x) moved…
      1. 5 up
      2. 5 to the right
      3. 5 to the left
   2. (3, 2) is on y = f(x). On y = f(x) − 4 it goes to (3, ___)
   3. (3, 2) is on y = f(x). On y = f(x + 1) it goes to (___, 2)
   4. y = x² moved 3 to the left is y = (x + ___)²
   5. y = x² is reflected in the y-axis, giving y = (−x)². What happens to the curve?
      1. Nothing: it is the same curve
      2. It turns upside down
      3. It moves to the left
   6. y = −f(x) is y = f(x) reflected in which axis?
      1. The x-axis
      2. The y-axis
   7. x² − 6x + 11 = (x − 3)² + 2, so y = x² − 6x + 11 is y = x² moved by the vector (___, ___)
   8. Stretch: y = f(x) has its lowest point at (1, −3). The lowest point of y = f(x − 2) + 5 is (___, ___)

   Support · For question 3: f(x + 1) is inside the bracket, so only x changes, the opposite way to the sign. Which way, and how far?

   For the teacher

   Everyone chooses before anyone explains. If they moved it the way the sign looks, ask where (x − 4)² is zero. If they moved it up or down, ask whether the number is inside or outside the bracket.

   Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7 links to completing the square. The stretch combines two moves.

   Students show their working on paper or the board; the quiz keeps their final answers.

   If many moved it the way the sign looks, ask where (x − 4)² is 0 before they practise.

   Shorter: After the vote, questions 1–4 only (−6 min).

   Longer: The stretch, then: does the order of two moves matter? (+10 min).

5. 5 · Show what you know5 min

   Describe the move

   y = x² moves to y = (x + 2)² − 1.

   The vector? Where does the lowest point go?

   Say why, in one sentence.

   Write here

   5 min 5 · Exit · students show

   Describe the move

   One description and one point show who reads inside and outside correctly.

   Say “Describe the move as a vector, and say where the lowest point goes.”

   Watch for A vector with the wrong sign for the move inside the bracket.

   A vector and one sentence are enough.

   1. y = x² is moved to y = (x + 2)² − 1. Describe the move as a vector, and say where the lowest point goes, and why.

   For the teacher

   y = x² is moved to y = (x + 2)² − 1. Describe the move as a vector.

   Then say where the lowest point goes, and why.

   The answers show who still moves the curve the way the sign looks: start the next lesson with the hook's vote again. Set the homework.

   Longer: Sketch both curves and label both lowest points (+5 min).

The answers, their slides and the mistake behind each wrong answer come with your copy. The worked examples above are open to everyone.

One-to-one: ask the student to predict each move aloud and say why, then check it by working out one point, such as where the lowest point goes.

Homework: moves and points

For each move, say whether x or y changes, and which way. Give translations as vectors.

1. Describe how y = x² moves to y = x² + 6.
2. Describe how y = x² moves to y = (x − 5)².
3. (4, −3) is on y = f(x). Where does it go on y = f(x) + 2, on y = f(x − 3), on y = −f(x) and on y = f(−x)?
4. Write the equation of y = x² moved by the vector (−2, 3).
5. Sketch y = x² − 4 and y = −(x² − 4) on one pair of axes. Where do they meet?
6. Complete the square for y = x² + 4x + 1, and describe the move from y = x².

Print the worksheet with working space and homework. Teacher answers come with your Library copy.