Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)
Published: October 2025
Category: Algebra | Functions & Graphs
🔹 What Is a Function?
A function is a rule that assigns to every input (x-value) exactly one output (y-value).
In notation: y=f(x)y = f(x)y=f(x)
The graph of a function is the set of all points (x,y)(x, y)(x,y) that satisfy this relationship.
It visually represents how the output changes as the input varies.
A function turns numbers into shapes.
The graph turns equations into intuition.
🔹 Understanding Function Graphs
Every graph reveals a story — about growth, symmetry, rate of change, and behavior.
| Function Type | Example | Key Features |
|---|---|---|
| Linear | y=2x+3y = 2x + 3y=2x+3 | Straight line (constant rate of change) |
| Quadratic | y=x2−4x+3y = x^2 – 4x + 3y=x2−4x+3 | Parabola (U-shaped, symmetry) |
| Cubic | y=x3y = x^3y=x3 | S-curve, changes direction once |
| Exponential | y=exy = e^xy=ex | Rapid growth, always positive |
| Trigonometric | y=sinxy = \sin xy=sinx | Periodic wave pattern |
| Reciprocal | y=1xy = \frac{1}{x}y=x1 | Two branches, asymptotes |
| Absolute Value | ( y = | x |
🔹 The Function Graph Test — Vertical Line Rule
A graph represents a function if and only if no vertical line crosses it more than once.
✅ Passes: y=x2y = x^2y=x2, y=exy = e^xy=ex
❌ Fails: x=y2x = y^2x=y2 (not a function, since one x maps to two y values)
🔹 Step-by-Step Example — Sketching y=x2−4x+3y = x^2 – 4x + 3y=x2−4x+3
1️⃣ Identify function type: Quadratic (a parabola).
2️⃣ Find key points:
- Vertex using formula x=−b2a=42=2x = -\frac{b}{2a} = \frac{4}{2} = 2x=−2ab=24=2
- Substitute x=2x = 2x=2: y=(2)2−4(2)+3=−1y = (2)^2 – 4(2) + 3 = -1y=(2)2−4(2)+3=−1.
→ Vertex at (2,−1)(2, -1)(2,−1).
3️⃣ Find x-intercepts:
x2−4x+3=0⇒(x−1)(x−3)=0⇒x=1,3x^2 – 4x + 3 = 0 \Rightarrow (x – 1)(x – 3) = 0 \Rightarrow x = 1, 3×2−4x+3=0⇒(x−1)(x−3)=0⇒x=1,3.
4️⃣ Find y-intercept: x=0⇒y=3x = 0 \Rightarrow y = 3x=0⇒y=3.
✅ Graph: Upward-opening parabola with vertex (2, −1) and symmetry about x=2x = 2x=2.
🔹 Features to Analyze in a Function Graph
| Property | What It Shows | Example |
|---|---|---|
| Intercepts | Where graph crosses axes | xxx- and yyy-intercepts |
| Symmetry | Even, odd, or none | x2x^2×2 (even), x3x^3×3 (odd) |
| Asymptotes | Lines graph approaches but never touches | y=1xy = \frac{1}{x}y=x1 has asymptotes at x = 0 and y = 0 |
| Domain & Range | Allowed x- and y-values | exe^xex: domain = ℝ, range = (0, ∞) |
| Intervals | Increasing, decreasing, stationary points | Derived via derivative f′(x)f'(x)f′(x) |
| Turning Points | Local max/min where slope = 0 | f′(x)=0f'(x) = 0f′(x)=0 |
🔹 Example — y=∣x−2∣+1y = |x – 2| + 1y=∣x−2∣+1
Transform base function y=∣x∣y = |x|y=∣x∣:
- Shift right 2 → (x−2)(x – 2)(x−2)
- Shift up 1 → +1+1+1
✅ Vertex at (2, 1); V-shape symmetric about x = 2.
🔹 Symmetry in Graphs
- Even Function: f(−x)=f(x)f(-x) = f(x)f(−x)=f(x) → symmetric about y-axis
e.g., y=x2,y=∣x∣,y=cosxy = x^2, y = |x|, y = \cos xy=x2,y=∣x∣,y=cosx - Odd Function: f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x) → symmetric about origin
e.g., y=x3,y=sinxy = x^3, y = \sin xy=x3,y=sinx
🔹 Behavior & Limits
For large |x| values, end behavior shows what happens “at infinity”:
- y=x2→+∞y = x^2 \to +\inftyy=x2→+∞ as x→±∞x \to ±\inftyx→±∞
- y=e−x→0y = e^{-x} \to 0y=e−x→0 as x→+∞x \to +\inftyx→+∞
Knowing limits helps in sketching without plotting many points.
🔹 Common Mistakes
- ❌ Confusing the effect of +/− signs in transformations.
- ❌ Missing intercepts or asymptotes.
- ❌ Drawing incorrect symmetry.
- ❌ Not labeling axes or scales consistently.
🔹 Advanced Insight — Function Composition
When one function is applied to another, y=f(g(x))y = f(g(x))y=f(g(x)), the graph’s shape becomes a composition of transformations.
For example: y=x2+1y = \sqrt{x^2 + 1}y=x2+1
is a vertical shift of y=∣x∣y = |x|y=∣x∣, smoothed near the origin.
Understanding how g(x)g(x)g(x) modifies f(x)f(x)f(x) is key for advanced graph sketching (IB HL, A Level P4, STEP).
🔹 Real-World Applications
- Physics: motion, potential energy curves, velocity-time graphs
- Economics: cost, revenue, demand curves
- Biology: growth and decay models
- Machine Learning: activation and loss functions
🌟 Why It Matters
A function’s graph is more than a drawing — it’s a story of change.
Learning to interpret and sketch graphs builds your ability to think visually, analytically, and symbolically — essential for higher-level math.
📘 Learn Beyond Plotting
At Math By Rishabh, you learn how to see mathematics.
In the Mathematics Elevate Mentorship, we teach:
✅ Graph sketching through transformation logic,
✅ Function behavior through calculus,
✅ Step-by-step graph interpretation for exams like IB, AP, A Level, and STEP.
🚀 Turn equations into intuition.
👉 Book your personalized mentorship session now at MathByRishabh.com
