1st Year Math Chapter 3 Exercise 3.1 Notes: Quadratic Functions — Graphing and Solving (Punjab Board 2026-27)

Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.

Chapter 3, Theory of Quadratic Functions, follows Functions and Graphs in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 3.1 introduces the quadratic function in standard and vertex form, its graph (the parabola), and the main methods for solving quadratic equations.

What Does This Exercise Cover?

This exercise connects the general function ideas from Chapter 2 to the specific case of quadratic functions — converting between forms, identifying the vertex and axis of symmetry, and solving ax² + bx + c = 0 using three standard methods.

Key Concepts

1. What Is a Quadratic Function?

A quadratic function has the standard form f(x) = ax² + bx + c, where a, b, c are real numbers and a ≠ 0. Its graph is always a U-shaped (or upside-down U) curve called a parabola.

2. Vertex Form

Completing the square converts the standard form into vertex form, f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola — its lowest point if a > 0, or highest point if a < 0.

3. Axis of Symmetry

Every parabola is symmetric about a vertical line through its vertex, called the axis of symmetry, given by x = h, or equivalently x = −b/2a using the standard form coefficients.

4. Maximum and Minimum Value

If a > 0, the parabola opens upward and the vertex gives the function’s minimum value. If a < 0, the parabola opens downward and the vertex gives the function’s maximum value.

5. Solving Quadratic Equations

Three standard methods are used to solve ax² + bx + c = 0:

• Factorization — writing the trinomial as a product of two linear factors and setting each to zero
• Completing the square — rewriting the equation as a perfect square to isolate x
• Quadratic formula — x = (−b ± √(b² − 4ac)) / 2a, which works for every quadratic equation

Choosing a Method to Solve ax² + bx + c = 0

Solved Examples

Example 1: Convert f(x) = x² − 6x + 5 into vertex form, and state the vertex.

Complete the square: x² − 6x + 5 = (x² − 6x + 9) − 9 + 5.

This simplifies to (x − 3)² − 4.

Answer: f(x) = (x − 3)² − 4, with vertex (3, −4)

Example 2: Solve x² − 5x + 6 = 0 by factorization.

Find two numbers that multiply to 6 and add to −5: these are −2 and −3.

Factor: (x − 2)(x − 3) = 0.

Set each factor to zero: x − 2 = 0 or x − 3 = 0.

Answer: x = 2 or x = 3

Example 3: Solve 2x² + 3x − 2 = 0 using the quadratic formula.

Here a = 2, b = 3, c = −2.

Discriminant: b² − 4ac = 9 − 4(2)(−2) = 9 + 16 = 25.

x = (−3 ± √25) / (2×2) = (−3 ± 5) / 4.

Answer: x = 1/2 or x = −2

Sample MCQs

1. The graph of a quadratic function is called a:

a) Straight line   b) Parabola   c) Circle   d) Hyperbola

Answer: b) Parabola

2. In vertex form f(x) = a(x − h)² + k, the vertex is located at:

a) (a, k)   b) (h, k)   c) (k, h)   d) (0, 0)

Answer: b) (h, k)

3. The axis of symmetry of f(x) = ax² + bx + c is given by:

a) x = b/2a   b) x = −b/2a   c) x = −b/a   d) x = c/a

Answer: b) x = −b/2a

4. If a < 0 in a quadratic function, the parabola:

a) Opens upward, has a minimum   b) Opens downward, has a maximum   c) Is a straight line   d) Has no vertex

Answer: b) Opens downward, has a maximum

5. Solving x² − 9 = 0 gives:

a) x = ±3   b) x = 9   c) x = 3 only   d) x = −9

Answer: a) x = ±3

Important Short Questions

• Define a quadratic function and state its standard form.
• What is meant by the vertex of a parabola?
• State the formula for the axis of symmetry of a quadratic function.
• Solve x² − 7x + 12 = 0 by factorization.
• Explain how the sign of a determines whether a quadratic function has a maximum or minimum value.

Important Long Questions

• Convert f(x) = 2x² − 8x + 3 into vertex form by completing the square, and state the vertex and axis of symmetry.
• Solve 3x² − 5x − 2 = 0 using the quadratic formula, showing all steps.
• Explain the relationship between the sign of a in f(x) = ax² + bx + c and both the direction the parabola opens and its maximum or minimum value.
• Find the vertex, axis of symmetry, and x-intercepts of f(x) = x² − 4x + 3, and describe its graph.

How to Approach This Exercise Effectively

1. Practice completing the square on several examples until converting standard form to vertex form feels automatic.
2. Memorize the quadratic formula exactly — a single sign error is the most common mistake in this exercise.
3. Always calculate the discriminant (b² − 4ac) before choosing a method — it often reveals whether the equation factors nicely.
4. Check factorization answers by expanding your two binomials back out to confirm they match the original trinomial.
5. Sketch a quick parabola for every problem involving a vertex or intercepts — visualizing the shape helps catch sign errors.

FAQs

Q: What is the difference between standard form and vertex form?

A: Standard form (ax² + bx + c) is easiest for identifying coefficients quickly, while vertex form (a(x−h)² + k) directly reveals the vertex and is more useful for graphing.

Q: Does every quadratic equation have real solutions?

A: No — if the discriminant (b² − 4ac) is negative, the equation has no real solutions, only complex ones (covered further in Exercise 3.2).

Q: Which solving method should I use first?

A: Try factorization first since it’s fastest when it works; if the trinomial doesn’t factor easily, move to the quadratic formula, which always works.

Q: Why does the sign of a affect whether the vertex is a maximum or minimum?

A: When a > 0, the parabola opens upward, so its vertex is the lowest point (minimum); when a < 0, it opens downward, making the vertex the highest point (maximum).

Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 3: Theory of Quadratic Functions, Exercise 3.1.