Graphs are the language of engineering analysis. Every graph type you need for T Level / Level 3 has a distinct shape and a distinct meaning. Learn to recognise, sketch, and interpret all six types — then apply them to motion, force, and electrical engineering problems.
Linear y=mx+cQuadratic y=ax²+bx+cCubic y=ax³+…Reciprocal y=k/xExponential y=aˣMotion graphs
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Linear Graphs — y = mx + c
A straight line. The gradient m tells you how steep it is and which way it slopes. The y-intercept c is where it crosses the y-axis. These are the most important graphs in engineering analysis.
y = mx + c m = gradient = Δy/Δx = rise/run c = y-intercept (value when x=0)
Gradient — four types
m > 0 → slopes upward (left to right)
m < 0 → slopes downward (left to right)
m = 0 → horizontal line y = c
m undefined → vertical line x = k
💡 Gradient from two points: m = (y₂−y₁)/(x₂−x₁). Always subtract in the same order.
A parabola — always U-shaped (a>0) or ∩-shaped (a<0). Has one turning point (the vertex), up to two roots, and a vertical axis of symmetry.
y = ax²+bx+c Vertex at x=−b/(2a) Roots: where y=0 y-intercept: y=c when x=0
Key features to identify
Shape: U if a>0 ∩ if a<0
y-intercept: substitute x=0 → y=c
Roots (x-intercepts): solve ax²+bx+c=0
Vertex x-coord: x=−b/(2a)
Axis of symmetry: vertical line x=−b/(2a)
💡 Completing the square gives the vertex directly: y=a(x−h)²+k → vertex at (h,k)
Worked examples
Sketch y=x²−5x+4. Find roots and vertex.
Factorise: (x−1)(x−4)=0
Roots: x=1 and x=4
Vertex x: −(−5)/(2×1)=2.5
Vertex y: 2.5²−5(2.5)+4=−2.25
Vertex:(2.5,−2.25) ✅
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Cubic Graphs — y = ax³ + bx² + cx + d
An S-shaped curve with up to two turning points and up to three roots. The sign of a determines which way the S goes.
a > 0: bottom-left to top-right (like ∫) a < 0: top-left to bottom-right (like S flipped)
Key cubic features
Up to 3 real roots (x-intercepts)
Exactly 2 turning points (or 0 if they coincide)
y-intercept: substitute x=0 → y=d
As x→+∞: y→+∞ if a>0, y→−∞ if a<0
As x→−∞: y→−∞ if a>0, y→+∞ if a<0
💡 Cubics in engineering often model stress-strain curves beyond the elastic limit, or cam follower displacement profiles.
Sketching a cubic
1️⃣ Find y-intercept (x=0)
2️⃣ Find x-intercepts (solve =0, or spot factors)
3️⃣ Identify sign of a → direction of curve
4️⃣ Find turning points (dy/dx=0 if using calculus)
5️⃣ Plot a few extra points to confirm shape
📌 At T Level you mainly need to recognise and read cubics, not differentiate them. Focus on roots, intercepts, and shape.
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Reciprocal Graphs — y = k/x
Two smooth curves (a hyperbola) that never touch the axes. The axes are asymptotes — lines the curve gets infinitely close to but never reaches.
y = k/x = kx⁻¹ Asymptotes at x=0 (y-axis) and y=0 (x-axis) As x→0⁺, y→+∞ (if k>0)
Key features
k > 0 → curves in Q1 (top-right) and Q3 (bottom-left)
k < 0 → curves in Q2 (top-left) and Q4 (bottom-right)
No x-intercepts or y-intercepts — ever
Never touches x=0 or y=0 (asymptotes)
Product xy = k always (constant product)
💡 Engineering: Boyle's Law PV=constant is a reciprocal graph (pressure vs volume at constant temperature). Ohm's Law I=V/R at constant V is also reciprocal (current vs resistance).
Reading from the graph
At x=1: y=k (gives you k directly)
At x=2: y=k/2 (half the value at x=1)
At x=0.5: y=2k (double the value at x=1)
y=k/x passes through (3,6). Find k and y at x=9.
Find k: 6=k/3 → k=18
At x=9: y=18/9=2 ✅
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Exponential Graphs — y = aˣ and y = e^x
The exponential graph models growth and decay — the most important curve in engineering, electronics and biology. It always crosses the y-axis at (0,1) and has the x-axis as an asymptote.
y = aˣ (a>1 → growth) y = a⁻ˣ (decay) y-intercept always (0,1) Never reaches y=0
Key properties
y-intercept: always (0,1) since a⁰=1
Growth (a>1): rises steeply to the right
Decay (0<a<1): falls toward x-axis from left
Asymptote: y=0 (x-axis) — never reaches zero
y=2⁻ˣ is the mirror of y=2ˣ in the y-axis
💡 e≈2.718 is the natural base. y=eˣ is the most common in engineering: Newton's law of cooling, capacitor charging, radioactive decay all use eˣ.
Growth and decay models
Growth: N = N₀ × eˡᵗ (population, bacteria)
Decay: N = N₀ × e⁻ˡᵗ (radioactive, cooling)
Half-life: t½ = ln2/λ ≈ 0.693/λ
Capacitor: V=12·e⁻²ᵗ. Find V at t=0.5s.
Substitute: V=12×e⁻²⁽⁰·⁵⁾=12×e⁻¹
e⁻¹=0.3679: V=12×0.3679
Answer:V=4.41V ✅
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Motion Graphs — Distance-Time and Velocity-Time
The two most important applied graphs in engineering. Everything comes from one rule: gradient = rate of change.
Distance-time: gradient = speed Velocity-time: gradient = acceleration area under graph = distance
Distance-time interpretation
What you see
What it means
Steep positive gradient
High speed
Horizontal line (zero gradient)
Stationary (stopped)
Negative gradient
Moving back towards start
Curved line
Changing speed (acceleration)
Speed = gradient = Δd/Δt
Velocity-time interpretation
What you see
What it means
Positive gradient
Acceleration
Negative gradient
Deceleration
Horizontal line
Constant velocity
Area under graph
Distance travelled
a = gradient = Δv/Δt d = area under v-t graph
From the v-t graph above: find acceleration and total distance
Acceleration: gradient in phase 1 = (15−0)/(4−0) = 3.75 m/s²
Distance (0→4s): triangle = ½×4×15 = 30 m
Distance (4→7s): rectangle = 3×15 = 45 m
Distance (7→10s): triangle = ½×3×15 = 22.5 m
Total distance:30+45+22.5 = 97.5 m ✅
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Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
🌡️Maintenance:Cooling curve: exponential graph, read off time constants.
🔧Testing:F–x graph: gradient = spring constant k.
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Engineering Applications of Graphs
Every graph type you've studied models a real engineering phenomenon. Knowing which graph fits which situation is a core T Level skill.
📈 Linear — Ohm's Law (V = IR)
Voltage vs current in a resistor gives a straight line. Gradient = resistance R (ohms). A steeper line means higher resistance.
V = IR → y = mx + 0 (through origin) Gradient m = R
Example: R=100Ω At I=0.1A: V=10V At I=0.5A: V=50V
Plot: x-axis = current (A) y-axis = voltage (V) gradient = 100 V/A = 100Ω ✓
Non-ohmic devices (diodes, LEDs): curve, not line
📈 Linear — Hooke's Law (F = kx)
Force vs extension in a spring is linear until the elastic limit. Gradient = spring constant k (N/m). Beyond the limit — graph curves (plastic deformation).
F = kx → straight line through origin k = gradient = F/x (N/m or N/mm)
Example: spring extends 12mm per 50N k = 50/0.012 = 4167 N/m
Elastic potential energy: E = ½kx² (area under F-x graph) = ½×4167×0.012² = 0.3 J
Graph becomes non-linear beyond elastic limit
∪ Quadratic — Projectile Motion
The path of a thrown object (ignoring air resistance) follows a quadratic curve. Height vs horizontal distance is a downward parabola.
h = h₀ + v₀t − ½gt² g = 9.81 m/s²
Example: launched at 20m/s vertically from 5m: h = 5 + 20t − 4.905t²
Max height: t = v₀/g = 20/9.81 = 2.04s h_max = 5+20(2.04)−4.905(2.04)² = 5+40.8−20.4 = 25.4m
Hits ground when h=0: solve quadratic
⤢ Reciprocal — Boyle's Law (PV = const)
Pressure vs volume for a gas at constant temperature is a reciprocal curve. Doubling volume halves pressure.
PV = k → P = k/V
Example: gas at P₁=200kPa, V₁=3m³ k = 200×3 = 600 kPa·m³
At V=6m³: P = 600/6 = 100kPa (halved) At V=1m³: P = 600/1 = 600kPa
Graph: P on y-axis, V on x-axis → reciprocal curve in Q1 only (pressure and volume both positive)
📉 Exponential — Capacitor Charging/Discharging
A capacitor charges and discharges exponentially. The time constant τ=RC determines how fast. This is one of the most important curves in electronics.
Discharging: V = V₀·e^(−t/RC) Charging: V = V₀(1−e^(−t/RC))
Time constant: τ = RC (seconds) At t=τ: V = 0.368·V₀ (discharging) At t=τ: V = 0.632·V₀ (charging) At t=5τ: fully discharged/charged (99.3%)
Example: R=10kΩ, C=100μF τ=10000×0.0001=1s At t=2s: V=12·e^(−2)=12×0.135=1.62V
🚗 Motion — CNC Machine Tool Path
A CNC cutting tool accelerates, runs at constant feedrate, then decelerates. The velocity-time graph determines material removal rate and surface finish.
Area under v-t graph = total tool travel This ensures accurate part length
〜 Cubic — Stress-Strain Beyond Elastic Limit
The stress-strain curve for mild steel has a cubic-like shape beyond yielding: linear elastic region, yield point, plastic region, then fracture.
Elastic region: σ = Eε (linear, Hooke's law) E = Young's modulus (gradient) Steel: E ≈ 200 GPa
Beyond yield: curve flattens (plastic flow) Approximated as cubic for modelling: σ = Eε − kε³ (some models)
Key readings from the graph: • Gradient of linear part = E • Yield stress: where graph first curves • Ultimate tensile strength: peak • Fracture point: where it ends
🔄 Gradient = Rate of Change
The single most important graph skill: the gradient of any graph tells you the rate of change of the quantity on the y-axis with respect to x.
Area under graph = accumulated quantity: v-t area = distance F-d area = work done (J) P-t area = energy (J) I-t area = charge (C)
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Interactive Graph Plotter
Plot any graph type by adjusting the parameters. Use the dropdown to switch between graph families, then drag the sliders or type values to see the curve update instantly.
📊 Graph Plotter — All TypesLive plotting with key features
ℹ️ How to use: Choose a graph type from the dropdown, then adjust the parameter sliders. The graph redraws instantly. Key features (roots, vertex, intercepts, asymptotes) are highlighted and listed below the graph.
Root / x-intercept: where the graph crosses the x-axis (y = 0)
y-intercept: where the graph crosses the y-axis (x = 0)
Vertex: the turning point of a parabola or cubic
Asymptote: a line the curve approaches but never reaches
Gradient reminder
Gradient m = rise / run = Δy / Δx
Positive m → slopes up left to right
Negative m → slopes down left to right
m = 0 → horizontal line | vertical → undefined
💡 For a straight line, gradient is the same at every point. For a curve, the gradient changes — it equals the slope of the tangent at that point.
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Quick Fire Quiz
Test your knowledge — 10 questions, instant feedback.
⚡ Graphs Blitz
ℹ️ How to play: A graph question appears on screen. Click the correct answer from the four options. Your score and streak update after each question — click Next to continue.
Score: 0Streak: 0 🔥Q 1/10
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Score: 0/10
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Practice Questions
Show all working. Always state which graph type applies and why.
🟢 Tier 1 — Graph Recognition and Reading
ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer to reveal the full worked solution and mark scheme.
Q1
State the gradient and y-intercept of: (a) y=4x−3 (b) y=−2x+7 (c) 3y=6x+9
[6]
(a) m=4, c=−3 (b) m=−2, c=7 (c) y=2x+3 → m=2, c=3
Q2
Find the equation of the line through (2,5) and (6,13).
[3]
m=(13−5)/(6−2)=8/4=2 5=2(2)+c → c=1 y=2x+1
Q3
For y=x²−2x−3: (a) find the roots, (b) find the vertex, (c) state the y-intercept.
y=k/x passes through (5,8). Find k. What is y when x=20?
[3]
k=5×8=40 At x=20: y=40/20=2
Q5
A v-t graph shows a vehicle accelerating from 0 to 24 m/s in 8s, then travelling at 24 m/s for 5s. Find (a) acceleration, (b) total distance.
[4]
(a) a=Δv/Δt=24/8=3 m/s² (b) Triangle=½×8×24=96m Rectangle=5×24=120m Total=216 m
🟡 Tier 2 — Engineering Contexts
Q6
A spring obeys F=kx. When F=60N, x=15mm. Find k. What force extends the spring 40mm?
[3]
k=F/x=60/0.015=4000 N/m F=4000×0.04=160 N
Q7
Gas at P=150kPa, V=4m³. Using Boyle's law PV=k, find pressure when V=2.5m³.
[3]
k=150×4=600 P=600/2.5=240 kPa
Q8
A capacitor discharges: V=20·e^(−t/3) volts. Find V at t=3s and t=9s. What is the time constant?
[4]
τ=RC=3s At t=3: V=20×e⁻¹=20×0.3679=7.36V At t=9: V=20×e⁻³=20×0.0498=0.996V
🔴 Tier 3 — T Level Challenge
Q9
A v-t graph has three phases: (i) linear increase 0→18m/s over 6s, (ii) constant at 18m/s for 10s, (iii) linear decrease to 0 over 4s. Find (a) acceleration in phase i, (b) deceleration in phase iii, (c) total distance, (d) average speed for the whole journey.
[6]
(a) a=18/6=3 m/s² (b) decel=18/4=4.5 m/s² (c) ½×6×18+10×18+½×4×18=54+180+36=270 m (d) avg=270/20=13.5 m/s
Q10
A current-voltage graph for a component shows a straight line through origin with gradient 0.02 A/V. (a) Identify the component type, (b) find the resistance, (c) find current at 240V, (d) find power at 240V using P=VI.
[5]
(a) Ohmic resistor (straight line through origin) (b) Gradient=I/V=1/R → R=1/0.02=50Ω (c) I=240/50=4.8 A (d) P=240×4.8=1152 W = 1.15 kW