How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
🔥
Warm-Up (Retrieval Practice)
💡 Quick Knowledge Check
Test your prior knowledge before diving in.
Q1 Write the coordinates of (−2, 4).
Q2 What shape is the graph of y = x²?
Q3 True or False: y = 1/x crosses the axes.
Q4 🔄 RECALL Solve: 4x − 3 = 13
Q4 Solve: 4x − 3 = 13
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
Big Idea: Curved Graphs
💡 The Core Concept
Not all graphs are straight lines. Some graphs curve because the rate of change is not constant. In this unit we study quadratic graphs (y = x² type) and reciprocal graphs (y = 1/x type).
Straight lines have constant slope — curved graphs change speed
📖
Core Definitions
Quadratic graph
A U-shaped curve (parabola) from equations like y = x², y = −x² + 3
Reciprocal graph
Two separate curves from y = k/x — gets close to axes but never touches them
Symmetry
When one half of the graph mirrors the other — quadratics are symmetrical about a vertical line
Asymptote
A line the graph approaches but never actually reaches or crosses
📐
Part 1: Quadratic Graphs
Worked Example 01Table of Values (y = x²)
Complete the table and plot y = x²
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = x² | 4 | 1 | 0 | 1 | 4 |
Plot the points and join with a smooth curve.
y = x² — U-shaped, symmetrical about the y-axis, lowest point at (0, 0)
✓ Answer
The graph is: U-shaped · Symmetrical about the y-axis · Lowest point at (0, 0)
📘 Why It's Symmetrical
For y = x²: (−2)² = 4 and 2² = 4.Positive and negative x-values give the same y-value. This creates a mirror image either side of the y-axis.
Worked Example 02Negative Quadratic (y = −x²)
Graph y = −x²
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = −x² | −4 | −1 | 0 | −1 | −4 |
y = −x² — Upside-down U, still symmetrical, highest point at (0, 0)
✓ Answer
Upside-down U · Still symmetrical · Highest point at (0, 0)
Worked Example 03Vertical Shift (y = x² + 2)
Graph y = x² + 2
Effect of the +2
This shifts the entire graph up 2 units. Every y-value is 2 more than in y = x².
Adding +2 lifts the whole curve up — the lowest point moves from (0, 0) to (0, 2)
✓ Answer
✓ Lowest point becomes (0, 2) — same U-shape, just shifted up
📐
Part 2: Reciprocal Graphs
Worked Example 04Table of Values (y = 1/x)
Complete the table and plot y = 1/x
| x | −2 | −1 | −0.5 | 0.5 | 1 | 2 |
|---|---|---|---|---|---|---|
| y = 1/x | −0.5 | −1 | −2 | 2 | 1 | 0.5 |
Plot points and draw two separate curves.
Notice: there is no value when x = 0 (you can't divide by zero).
y = 1/x — Two curves in Quadrants I and III. Both axes are asymptotes (graph never touches them).
✓ Answer
Graph never touches x-axis · Graph never touches y-axis · Both axes are asymptotes
📌 The graph gets closer and closer to the axes but never actually reaches them. That's what "asymptote" means.
Worked Example 05Negative k in Reciprocal (y = −2/x)NEW
Graph y = −2/x
| x | −2 | −1 | 1 | 2 |
|---|---|---|---|---|
| y = −2/x | 1 | 2 | −2 | −1 |
Key observation:
When x is positive, y is negative. When x is negative, y is positive.
This flips the curves to the opposite quadrants.
This flips the curves to the opposite quadrants.
y = −2/x — Curves in Quadrants II and IV (opposite to y = 1/x). Axes are still asymptotes.
✓ Answer
Graph lies in Quadrant II and Quadrant IV · The axes remain asymptotes
📐 Part 3 — Using Graphs to Solve Equations
Examiners love asking you to read solutions from a graph. This skill connects graphing to algebra.Worked Example 06Solving from a Quadratic GraphEXAM STYLE
The graph of y = x² − 4 is shown. Use it to solve x² − 4 = 0.
1
"= 0" means
y = 0, so look where the curve crosses the x-axis.2
Read the x-values at the crossing points from the graph.
The curve y = x² − 4 crosses the x-axis at x = −2 and x = 2
✓ Solutions
x = −2 or x = 2
⚠️ Quadratic equations can have two solutions — always check both sides of the curve!
✅
Process Checklists
✅ Quadratic Graphs
✓
Use a table of values✓
Plot points accurately✓
Join with smooth curve✓
Identify symmetry✓
Never use straight lines✅ Reciprocal Graphs
✓
Use a table of values✓
Plot points carefully✓
Draw two separate curves✓
Graph approaches but never touches axes✓
Check which quadrants🧮
Arithmetic Support Strip(−3)² = 9
(−3)³ = −27
1 ÷ negative = negative
negative ÷ positive = negative
−(−) = positive
x = 0 → undefined for 1/x
✏️
Practice Questions💪 Three-Tier Practice
Progress through Core → Multi-Step → Exam-Style. Tap any card to reveal working.🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Evaluate: (−3)²
✍️ Type Your Answer
What is (−3)²?
(−3) × (−3) = 9 (negative × negative = positive)
✓ 9
Problem 1
Complete table for y = x² (x = −1, 0, 1).
x = −1: y = (−1)² = 1
x = 0: y = 0² = 0
x = 1: y = 1² = 1
x = 0: y = 0² = 0
x = 1: y = 1² = 1
✓ y-values are 1, 0, 1
👆 Tap to reveal
Problem 2
State the shape of y = −x².
The negative flips the parabola upside down.
✓ Upside-down U (inverted parabola)
👆 Tap to reveal
Problem 3
Complete table for y = 1/x (x = 1, 2).
x = 1: y = 1/1 = 1
x = 2: y = 1/2 = 0.5
x = 2: y = 1/2 = 0.5
✓ y-values are 1, 0.5
👆 Tap to reveal
🟠 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
What is y when x = 2 for y = x² + 1?
Problem 4
Plot y = x² − 1.
x = −2: y = 4 − 1 = 3
x = −1: y = 1 − 1 = 0
x = 0: y = 0 − 1 = −1
x = 1: y = 1 − 1 = 0
x = 2: y = 4 − 1 = 3
x = −1: y = 1 − 1 = 0
x = 0: y = 0 − 1 = −1
x = 1: y = 1 − 1 = 0
x = 2: y = 4 − 1 = 3
✓ U-shaped curve, lowest point (0, −1)
👆 Tap to reveal
Problem 5
Plot y = −x² + 3.
x = −2: y = −4 + 3 = −1
x = −1: y = −1 + 3 = 2
x = 0: y = 0 + 3 = 3
x = 1: y = −1 + 3 = 2
x = 2: y = −4 + 3 = −1
x = −1: y = −1 + 3 = 2
x = 0: y = 0 + 3 = 3
x = 1: y = −1 + 3 = 2
x = 2: y = −4 + 3 = −1
✓ Inverted U-curve, peak at (0, 3)
👆 Tap to reveal
Problem 6
Complete table for y = 2/x.
x = −2: y = 2/(−2) = −1
x = −1: y = 2/(−1) = −2
x = 1: y = 2/1 = 2
x = 2: y = 2/2 = 1
x = −1: y = 2/(−1) = −2
x = 1: y = 2/1 = 2
x = 2: y = 2/2 = 1
✓ y-values are −1, −2, 2, 1 (same shape as y = 1/x, Q I & III)
👆 Tap to reveal
Problem 7
Complete table for y = −1/x.
x = −2: y = −1/(−2) = 0.5
x = −1: y = −1/(−1) = 1
x = 1: y = −1/1 = −1
x = 2: y = −1/2 = −0.5
x = −1: y = −1/(−1) = 1
x = 1: y = −1/1 = −1
x = 2: y = −1/2 = −0.5
✓ y-values are 0.5, 1, −1, −0.5 (Q II & IV like y = −2/x)
👆 Tap to reveal
🔴 Red — Exam Style
🔴 Red · TYPE YOUR ANSWER
What is y when x = −2 for y = 1/x?
Problem 8 · 2 marks
Sketch y = x² + 3.
x = −2: y = 7 | x = −1: y = 4 | x = 0: y = 3
x = 1: y = 4 | x = 2: y = 7
x = 1: y = 4 | x = 2: y = 7
✓ U-curve shifted up 3 units, vertex at (0, 3)
👆 Tap to reveal
Problem 9 · 2 marks
Sketch y = −x².
x = −2: y = −4 | x = −1: y = −1 | x = 0: y = 0
x = 1: y = −1 | x = 2: y = −4
x = 1: y = −1 | x = 2: y = −4
✓ Inverted U-shape, peak at (0, 0)
👆 Tap to reveal
Problem 10 · 2 marks
Sketch y = −2/x.
x = −2: y = 1 | x = −1: y = 2
x = 1: y = −2 | x = 2: y = −1
x = 1: y = −2 | x = 2: y = −1
✓ Reciprocal in Q II & IV, never touches axes
👆 Tap to reveal
0 / 10
Problems revealed
Tap problems to reveal working and answers
🔍
Mistake Detective⚠️ These mistakes cost marks
Study each one carefully.❌ Wrong
Joining quadratic points with straight lines
Quadratics must be smooth curves!
✅ Correct
Smooth, curved line connecting all points
❌ Wrong
Drawing reciprocal graph crossing the axes
Reciprocal graphs NEVER touch the axes
✅ Correct
Curves approach axes asymptotically
❌ Wrong
y = −2/x in Quadrants I & III
Negative k flips to opposite quadrants
✅ Correct
y = −2/x lies in Quadrants II & IV
(positive k → Q I & III, negative k → Q II & IV)
(positive k → Q I & III, negative k → Q II & IV)
🎓
Examiner Lens🎓 To Gain Full Marks
- Use a table of values (minimum 4–5 points)
- Plot at least 4 points accurately
- Draw smooth curve — never straight lines for quadratics
- Identify symmetry clearly
- Show asymptotes clearly for reciprocals (dashed lines along axes)
- Do not extend reciprocal curves through the axes
🏆
Mastery Checklist🏆 Unit 7 Mastery
✓
I can sketch y = x²
✓
I can sketch y = −x²
✓
I understand vertical shifts in quadratic graphs
✓
I can sketch y = 1/x and identify asymptotes
✓
I can sketch y = −k/x in the correct quadrants
✓
I understand asymptotes and how graphs approach them
Nice work — you’ve completed Unit 7!
3 units remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.
Ready for Unit 8?
If you can do these 3 things confidently, you're ready to move on:
👨👩👧 For Parents
Your child has completed Unit 7 of a structured GCSE Foundation Maths revision programme covering Algebra and Graphs — key topics on every Foundation paper.
✔ AQA, Edexcel & OCR aligned
✔ 170+ practice questions
✔ Step-by-step explanations
Up Next
Unit 8: Real-Life Graphs →
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