Unit 1: Class 12 Math PECTAA Notes New Book

Introduction

Unit 1 of the Class 12 Mathematics PECTAA New Book lays the foundation for the entire calculus and function-based portion of the syllabus. It teaches students how to classify different types of functions and how to sketch, transform, and interpret their graphs skills that are used repeatedly in later units on differentiation, integration, and analytical geometry.

What This Unit Covers

Unit 1 is organized into five graded exercises (1.1 to 1.5), moving from basic graph sketching to real-life applications:

1. Graphs of Algebraic Functions Students learn to sketch polynomial functions given in factored form, such as quadratics written as f(x) = a(x – h)(x – k). The method involves locating x-intercepts, finding the vertex, checking end behaviour as x → ±∞, and plotting the y-intercept to build an accurate parabola or higher-degree curve.

2. Classification of Functions This section distinguishes algebraic functions (polynomials, rational functions, radicals) from transcendental functions (exponential, logarithmic, trigonometric, and their inverses). It further splits transcendental functions into fundamental (e.g. aˣ, ln x, sin x) and non-fundamental types (combinations like e^(x²) or sin x + ln x).

3. Exponential and Hyperbolic Functions Students study the exponential function f(x) = aˣ, its domain, range, and graph shape for different bases, along with the natural exponential function eˣ. This section also introduces hyperbolic functions (sinh, cosh, tanh) and their identities, plus inverse hyperbolic formulas expressed in logarithmic form.

4. Logarithmic Functions This part covers the graph of y = logₐ x, its domain (x > 0), vertical asymptote at x = 0, fixed x-intercept at (1, 0), and continuity. Graphs are compared for bases greater than 1 versus bases between 0 and 1.

5. Inverse Functions and Transformations Students verify whether a function is one-to-one using the horizontal line test, find inverse functions algebraically, and confirm that a function and its inverse are reflections of each other about the line y = x. This section also covers transformations — vertical/horizontal shifts, stretches, compress ions, and combinations of these applied in a specific order.

6. Solving Exponential and Logarithmic Equations/Inequalities Students solve equations and inequalities involving exponential and logarithmic expressions, including cases requiring domain restriction and compatibility checks with the original expression.

7. Real-Life Applications The unit closes with applied problems: population growth and radioactive decay modelled by N = N₀e^(kt), sound intensity measured in decibels, and compound interest (both annual and continuous compounding).

Why This Unit Matters

Graphical Representation of Functions builds the visual and conceptual base for every later unit. Understanding how a graph shifts, stretches, or reflects, and being comfortable moving between a function and its inverse, makes topics like Further Differentiation (Unit 2) and Integration (Unit 3) far easier to follow. Board exams typically include at least one long question on graph sketching or transformations, and short questions from function classification and equation-solving are common.

How to Study Unit 1

• Master the factored-form method for sketching polynomial graphs before moving to transformations.
• Keep a quick-reference list of the standard graphs: y = xⁿ, y = aˣ, y = logₐ x, y = |x|, y = 1/x — most transformation questions are variations of these.
• Practise the horizontal line test and inverse-function steps together; they are usually tested as one combined question.
• For applied questions (growth, decay, decibels, compound interest), memorize which formula applies to which keyword (“compounded continuously” → e^(rt); “decibels” → 10 log(I/I₀)).

Practice Questions – Unit 1

Short Questions

1. Classify each of the following as algebraic or transcendental: f(x) = (2x + 1)⁴, g(x) = cos x + x², h(x) = √(x – 3), k(x) = 5ˣ.
2. State whether y = 7ˣ is a fundamental or non-fundamental transcendental function, and justify your answer.
3. Find the x-intercept(s) and vertex of f(x) = 3(x + 2)(x – 4).
4. Write the domain and range of f(x) = logₐ x for 0 < a < 1.
5. State the effect on the graph of y = f(x) when it is transformed to y = f(x – 3) + 2.
6. Use the horizontal line test to determine whether f(x) = x³ + 1 is one-to-one.

Long Questions

7. Sketch the graph of f(x) = 2(x + 1)(x – 3), clearly showing the x-intercepts, y-intercept, and vertex.
8. Given f(x) = √(x – 4), find the equation of the function after (i) a vertical stretch by factor 3, (ii) a horizontal shift 2 units to the left, (iii) applying both transformations in that order.
9. Let f(x) = log₂(x + 3). Find f⁻¹(x) algebraically, state the domain and range of both f and f⁻¹, and verify that their graphs are symmetric about y = x.
10. Solve the equation: 2^(x+1) – 3·2^(-x) = 5.
11. Solve the inequality: log₃(x – 1) ≤ log₃(7 – x).

Applied Questions

12. A bacterial culture grows according to N = 200e^(kt). If the population reaches 500 after 6 hours, find the growth constant k.
13. A radioactive isotope decays as N = 1000e^(-0.04t), where t is in years. Find the time taken for the amount to reduce to 250.
14. A sound has intensity I = 10⁻⁴ W/m² with reference intensity I₀ = 10⁻¹² W/m². Find its sound level in decibels.
15. A sum of Rs. 8000 is invested at 5% per annum, compounded continuously, for 5 years. Find the final amount.

These practice questions are written to reinforce Unit 1 concepts and are separate from the textbook’s own exercises — use them for extra revision before attempting the board-style exercises in the unit.

About HSA Notes

HSA Notes is an educational resource platform providing free study materials for Pakistani students, covering FBISE, Punjab, Sindh, and KP Boards in line with the New Books System. Materials span grades 9–12 across Mathematics, Chemistry, Biology, Physics, Computer Science, English, and Urdu, all prepared in a consistent, easy-to-follow branded format.

Besides free unit-wise notes and solved exercises, HSA Notes also offers online classes for students who want structured, teacher-guided sessions alongside self-study useful for learners who need concepts explained step by step or want doubt-solving support beyond what a PDF can provide.

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Conclusion

Unit 1 – Graphical Representation of Functions sets up the visual and conceptual tools that carry through the rest of Class 12 Mathematics. Mastering graph sketching, function classification, inverse functions, transformations, and the real-life applications of exponential and logarithmic functions gives students a strong base for Units 2 and 3, and a reliable set of marks in the board exam. Practising the solved examples alongside the questions above, and joining HSA Notes’ online classes for extra guidance where needed, will help students approach Unit 1 with confidence.