At a glance. One lesson of about 60 minutes. It assumes the vertex form (Lesson 3) and function notation (Lesson 1). Students apply translations, reflections and dilations to a point and to a graph, and write the rule that gives a described transformation.
Calculator: GDC optional. Everything can be done by hand. The optional section shows a GDC drawing a function and its image, and reminds students to read the rule rather than trust one sketch.
What students will learn
Know
- f(x − h) + k translates by (h, k).
- −f(x) and f(−x) reflect in the x-axis and y-axis.
- af(x) stretches vertically by a, and f(bx) squashes horizontally by 1 ÷ b.
Understand
- Outside the bracket acts on y, inside acts on x and does the opposite.
- A rule that holds for any f is a rule about the plane, not about a curve.
- A tracked point makes each rule checkable.
Do
- Find the image of a point under a transformation of f.
- Describe a transformation from its rule.
- Write the rule for a described transformation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the rules to points and features (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if and the figure. Students vary one parameter at a time and generalize the rule.
- Criterion C. Communicating: describing a transformation in words and in notation (worked examples; exit ticket).
- Criterion D. Applying mathematics in real-life contexts: the game ridge, where copies are made by changing a rule.
The one thing to do with the figure
Ask for a prediction before every move. With the identity on the screen, ask: “If I slide h to 3, which way does the ridge go?” Many say left because of the bracket; let the screen decide.
Then set h = 4 with the y-axis flip on and ask which was done first, the flip or the move. The followed corner settles it: (−2, 3) goes to (6, 3), so the flip came first. Leave the other order to the what-if and the next lesson.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the ridge. Collect predictions for f(x + 3) or f(x − 3). |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to inside and outside. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | (3, 5); 7; (1, 1). |
|---|---|
| Hook | f(x − 3): the ridge moves 3 to the right. |
| Worked examples | (1, 2); (−4, −1); the zero is at x = 2. |
| Faded example | (1, 11). |
| 1. Translation of a point | D. A moves left; C moves down; B reverses both. |
| 2. Horizontal translation of a point | −5. −1 − 4. |
| 3. Two reflections | −4. (3, −7). |
| 4. Horizontal squash | 2.5. 2x = 5. |
| 5. Stretch then shift of a maximum | 7. 3 × 4 − 5. |
| 6. Equation of a transformed parabola | −1. −9 + 8. |
| 7. Zeros of f(c − x) | 5. 3 − x = −2. |
| 8. Describing a transformation | C. A uses f(−x); B moves right; D moves up. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Moved the wrong way, reversed both, or reversed the y move. | Inside moves the opposite way; outside moves as written. |
| 3 or −4 in question 2 | Moved right, or solved x + 4 = 0. | Ask: where does the point go, not where does the bracket vanish? |
| 10, −10 or 4 in question 3 | Applied one reflection, or none. | Write the image coordinates before adding. |
| 10, 7 or 3 in question 4 | Doubled the zero, or shifted it. | Solve 2x = 5. |
| −3, 12 or −1 in question 5 | Put the 5 inside the 3, forgot it, or forgot the 3. | Apply the 3 and then the −5. |
| 17 or −17 in question 6 | Forgot the reflection, or reflected after the 8. | Write the rule in order. |
| 9, 1 or −3 in question 7 | Treated f(3 − x) as f(x − 3), or gave the smaller zero. | Solve 3 − x = each old zero. |
| A, B or D in question 8 | Confused f(−x) with −f(x), or inside and outside. | Test the point (2, 3). |
Questions worth asking
- Why does f(x − 3) move right? The output at x is what f gave at x − 3, so each feature of f appears 3 later.
- Which transformations change only the y-coordinates? Those outside the bracket: +k, −f, af.
- Does f(2x) stretch or squash? Squash: the x-coordinates are divided by 2.
- Is a reflection a dilation? It acts like a stretch with factor −1 in one direction, but in an exam call it a reflection.
Differentiation
A way in for anyone stuck
- Use a table with columns: rule, inside or outside, effect.
- Give a zigzag on a grid and let students plot the image of each corner first.
- Colour inside rules blue and outside rules red.
Stretch
- Describe fully the transformation that maps y = x² to y = (2x − 3)² − 1, as a translation after a dilation.
- Investigate: for which functions f is f(−x) = −f(x) for every x? What is the symmetry of the graph?
Students learning in English as an additional language
“Translate” means slide without turning. “Reflect” means flip as in a mirror. “Dilate” means stretch or squash. “Image” is the new point or graph after a transformation. Use the frames in the starters.
| Word | Plain meaning | Example |
|---|---|---|
| translation | sliding every point the same distance in the same direction | y = f(x) + 9 moves the graph 9 up |
| reflection | flipping a graph in a mirror line | y = −f(x) in the x-axis |
| dilation | stretching or squashing by a scale factor | y = 10f(x) |
| image | the point or graph after a transformation | (2, 8) is the image of (2, 3) under y = f(x) + 5 |
| scale factor | the number a length is multiplied by | 1 ÷ 2 for f(2x) |
| parent graph | the original graph before it is transformed | y = f(x) |
Sentence starters for explaining aloud or in writing:
- The graph is translated … units to the …
- The graph is dilated with scale factor … parallel to the … axis.
- The bracket is inside, so it acts on …
Two things not to say
“f(x + 3) moves right because the number is positive.” Inside the bracket it is the opposite: left.
“f(2x) stretches the graph.” It squashes it horizontally by a factor of 1 ÷ 2.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- TranslateThe point (2, 3) is on f. Where is its image on y = f(x − 4)?
(6, 3). - ReflectThe point (5, −2) is on f. Where is its image on y = −f(x)?
(5, 2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA zigzag path in a game has a corner at (6, 2). A copy is drawn by y = f(x + 5) + 3. Where is the copy's corner?
(1, 5). - Earlier work (Lesson 3)Write the vertex of y = −2(x + 1)² + 4.
(−1, 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe graph of f has a zero at x = −4. Find the zero of y = f(−2x).
x = 2. - CombinationWrite the rule for the graph of y = f(x) reflected in the y-axis and then translated 3 up.
y = f(−x) + 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.