Functions & Graphs | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Identify and sketch linear and quadratic graphsKenal pasti dan lukis graf linear dan kuadratik

Interpret distance-time graphs (gradient = speed)Tafsir graf jarak-masa (kecerunan = kelajuan)

Interpret speed-time graphs (gradient = acceleration, area = distance)Tafsir graf kelajuan-masa (kecerunan = pecutan, luas = jarak)

Calculate average speed and accelerationKira kelajuan purata dan pecutan

Solve problems involving motion graphsSelesaikan masalah melibatkan graf gerakan

📖 Key Concepts📖 Konsep Utama

Linear Functions (Form 3)Fungsi Linear (Tingkatan 3)

Straight line graphs of the form y = mx + c.Graf garis lurus berbentuk y = mx + c.

y = mx + c
m = gradient (slope), c = y-intercept
Positive m: line goes up ↗ | Negative m: line goes down ↘

Quadratic Functions (Form 3)Fungsi Kuadratik (Tingkatan 3)

Parabola graphs. The vertex is the maximum or minimum point.Graf parabola. Puncak ialah titik maksimum atau minimum.

y = ax² + bx + c
Vertex: x = −b/(2a)
If a > 0: minimum point (∪) | If a < 0: maximum point (∩)

Distance-Time Graphs (Form 4)Graf Jarak-Masa (Tingkatan 4)

Shows how distance changes over time. Gradient represents speed.Tunjukkan perubahan jarak dari masa ke masa. Kecerunan mewakili kelajuan.

Gradient = speed
Steeper slope = faster speed
Horizontal line = at rest

Speed-Time Graphs (Form 5)Graf Kelajuan-Masa (Tingkatan 5)

Shows how speed changes over time. Gradient = acceleration. Area = distance.Tunjukkan perubahan kelajuan dari masa ke masa. Kecerunan = pecutan. Luas = jarak.

🟣 Triangle area = ½ × base × height → distance during acceleration
🟢 Rectangle area = width × height → distance during constant speed🟣 Luas segitiga = ½ × tapak × tinggi → jarak semasa pecutan
🟢 Luas segi empat = lebar × tinggi → jarak semasa kelajuan malar

Gradient = acceleration
Area under graph = total distance
Triangle: ½ × base × height

Average Speed (Form 4)Kelajuan Purata (Tingkatan 4)

Total distance divided by total time taken.Jumlah jarak dibahagi dengan jumlah masa yang diambil.

Average speed = total distance / total time
⚠️ NOT the average of two speeds!

✏️ Worked Examples✏️ Contoh Penyelesaian

Draw a distance-time graph: Car travels 20 km in 30 min, stops 10 min, then 10 km in 15 min.Lukis graf jarak-masa: Kereta melaju 20km dalam 30min, berhenti 10min, kemudian 10km dalam 15min.

Plot points: (0,0), (0.5h, 20km), (1h, 20km), (1.25h, 30km)

Horizontal line during stop (gradient = 0 = at rest)

First leg: speed = 20/0.5 = 40 km/h, Second: 10/0.25 = 40 km/h

Graph shows two sloped segments at 40 km/h with flat rest period

Calculate acceleration: Speed increases from 0 to 20 m/s in 5 secondsKira pecutan: Kelajuan meningkat dari 0 kepada 20 m/s dalam 5 saat

Acceleration = change in speed / time

a = (20 − 0) / 5 = 4 m/s²

Gradient of speed-time graph = 4

Acceleration = 4 m/s²

Find distance from area under speed-time graphCari jarak dari luas di bawah graf kelajuan-masa

Car accelerates from rest at 2 m/s² for 10s, then constant 20 m/s for 5s

First: area = ½ × 10 × 20 = 100 m (triangle)

Second: area = 20 × 5 = 100 m (rectangle)

Total distance = 100 + 100 = 200 m

Total distance = 200 m

Find vertex of y = x² − 4x + 3Cari puncak bagi y = x² − 4x + 3

x = −b/(2a) = 4/2 = 2

y = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1

Since a = 1 > 0, this is a minimum point

Vertex at (2, −1) — minimum point

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Confusing gradient meaning between graph typesMengelirukan makna kecerunan antara jenis graf

In distance-time: gradient = speed. In speed-time: gradient = acceleration. These are DIFFERENT!Dalam jarak-masa: kecerunan = kelajuan. Dalam kelajuan-masa: kecerunan = pecutan. Ini BERBEZA!

❌ Forgetting area under speed-time graph equals distanceLupa luas di bawah graf kelajuan-masa sama dengan jarak

The area under a speed-time graph gives total distance. Split into triangles and rectangles if needed.Luas di bawah graf kelajuan-masa memberikan jumlah jarak. Bahagikan kepada segitiga dan segi empat tepat jika perlu.

❌ Wrong calculation of average speedPengiraan kelajuan purata yang salah

Average speed = total distance / total time. NOT the average of two speeds!Kelajuan purata = jarak keseluruhan / masa keseluruhan. Bukan purata dua kelajuan!

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

On a distance-time graph, gradient represents:Pada graf jarak-masa, kecerunan mewakili:

Speed

Acceleration

Distance

Time

In a speed-time graph, area under the curve gives:Dalam graf kelajuan-masa, luas di bawah lengkungan memberikan:

Speed

Distance

Acceleration

Time

Average speed = total distance / total time. True or false?Kelajuan purata = jarak keseluruhan / masa keseluruhan. Benar atau salah?

True

False

If speed increases uniformly, the speed-time graph is:Jika kelajuan meningkat seragam, graf kelajuan-masa ialah:

Horizontal

Straight line with positive gradient

Curved

Vertical

Quadratic function y = x² − 4x + 3, vertex at:Fungsi kuadratik y = x² − 4x + 3, puncak pada:

(2, −1)

(2, 3)

(0, 3)

(4, −1)

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

5 marks5 markah

A cyclist travels at constant speed for 2 hours covering 40 km, then rests for 1 hour, then cycles 20 km in 30 minutes.
(a) Draw the distance-time graph. [2 marks]
(b) Find the average speed for the whole journey. [3 marks]Penunggang basikal melaju pada kelajuan malar selama 2 jam meliputi 40 km, berehat 1 jam, kemudian berbasikal 20 km dalam 30 minit.
(a) Lukis graf jarak-masa. [2 markah]
(b) Cari kelajuan purata keseluruhan perjalanan. [3 markah]

Solution (a)Penyelesaian (a)

Plot (0,0) → (2,40): slope = 20 km/h

(2,40) → (3,40): horizontal (rest)

(3,40) → (3.5,60): slope = 40 km/h

Graph with two sloped segments and flat rest [2 marks]

Solution (b)Penyelesaian (b)

Total distance = 40 + 20 = 60 km

Total time = 2 + 1 + 0.5 = 3.5 hours

Average speed = 60 / 3.5 ≈ 17.14 km/h

Average speed ≈ 17.1 km/h [3 marks]

6 marks6 markah

A car accelerates from rest at 2 m/s² for 10 seconds, then travels at constant speed for 5 seconds.
(a) Find the maximum speed. [2 marks]
(b) Find the total distance travelled. [4 marks]Kereta memecut dari rehat pada 2 m/s² selama 10 saat, kemudian bergerak pada kelajuan malar selama 5 saat.
(a) Cari kelajuan maksimum. [2 markah]
(b) Cari jarak keseluruhan yang dilalui. [4 markah]

Solution (a)Penyelesaian (a)

v = u + at = 0 + 2(10) = 20 m/s

Maximum speed at end of acceleration phase

Maximum speed = 20 m/s [2 marks]

Solution (b)Penyelesaian (b)

Phase 1 (triangle): ½ × 10 × 20 = 100 m

Phase 2 (rectangle): 20 × 5 = 100 m

Total = 100 + 100 = 200 m

Total distance = 200 m [4 marks]

📋 Formula Summary📋 Ringkasan Formula

Functions & Graphs:Fungsi & Graf:

Linear: y = mx + c
Quadratic: y = ax² + bx + c, Vertex: x = −b/(2a)
Distance-time gradient = speed
Speed-time gradient = acceleration
Area under speed-time = distance
Average speed = total distance / total time