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Quadratic Functions (Forms 5–6)

Explores solving quadratic equations, completing the square, and analyzing graphs of quadratic functions.

📘 Topic Summary

Quadratic functions are a fundamental concept in Additional Mathematics, allowing us to model and analyze real-world phenomena. This study guide will explore the key concepts of solving quadratic equations, completing the square, and analyzing graphs of quadratic functions.

📖 Glossary
  • Quadratic Equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Completing the Square: A method for solving quadratic equations by manipulating the equation into the form (x - h)^2 = k.
  • Vertex Form: The standard form of a quadratic function in the form f(x) = a(x - h)^2 + k, where a is the leading coefficient and (h, k) is the vertex.
  • Axis of Symmetry: The vertical line that passes through the vertex of a parabola.
  • X-Intercept: The point at which a quadratic function intersects the x-axis.
⭐ Key Points
  • Quadratic equations can be solved using various methods, including factoring and completing the square.
  • Completing the square is a powerful technique for solving quadratic equations that are not easily factorable.
  • Vertex form is useful for analyzing the graph of a quadratic function, as it allows us to identify the vertex and axis of symmetry.
  • Quadratic functions can be used to model real-world phenomena, such as projectile motion and optimization problems.
  • The x-intercepts of a quadratic function represent the solutions to the equation.
  • The vertex of a quadratic function represents the maximum or minimum value of the function.
🔍 Subtopics
Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. For example, the equation x^2 + 5x + 6 = 0 is a quadratic equation. Solving these equations involves finding the values of x that make the equation true.

Solving Quadratic Equations by Factoring

Factoring is a method for solving quadratic equations where the left-hand side is rewritten as a product of two binomials. For example, to solve the equation x^2 + 5x + 6 = 0, we can rewrite it as (x + 3)(x + 2) = 0 and then set each factor equal to zero to find the solutions.

Completing the Square

Completing the square is a method for solving quadratic equations where the left-hand side is rewritten in the form (x + h)^2 + k. This can be done by taking half of the coefficient of the x term, squaring it, and adding it to both sides of the equation.

Vertex Form and Graph Analysis

The vertex form of a quadratic function is y = a(x - h)^2 + k. The graph of a quadratic function is a parabola that opens upward if 'a' is positive, downward if 'a' is negative, and has its vertex at the point (h, k).

Quadratic Functions in Real-World Applications

Quadratic functions are used to model many real-world phenomena, such as projectile motion, electrical circuits, and optimization problems. For example, the trajectory of a thrown ball can be modeled using a quadratic function.

Common Quadratic Equations

Some common quadratic equations include x^2 + 4x + 3 = 0, x^2 - 7x + 12 = 0, and x^2 + 2x + 1 = 0. These equations can be solved using various methods such as factoring or completing the square.

Quadratic Inequalities

Quadratic inequalities are inequalities involving quadratic expressions. For example, the inequality x^2 + 3x - 4 > 0 is a quadratic inequality. Solving these inequalities involves finding the intervals where the inequality is true.

Quadratic Functions in Additional Mathematics

Quadratic functions are used extensively in additional mathematics, particularly in calculus and algebra. They are used to model complex phenomena and solve problems that involve optimization and motion.

Practice Problems and Exercises

Solve the following quadratic equations: x^2 + 5x + 6 = 0, x^2 - 3x - 2 = 0. Solve the following quadratic inequality: x^2 + 2x - 3 > 0.

Common Mistakes and Errors

Some common mistakes when solving quadratic equations include forgetting to check for extraneous solutions, not simplifying the equation enough, or using an incorrect method. To avoid these errors, it is important to carefully read the problem, choose the correct method, and simplify the equation thoroughly.

Quadratic Functions in Other Subjects

Quadratic functions are used in many other subjects, including physics, engineering, and computer science. They are used to model complex phenomena such as motion, force, and energy.

🧠 Practice Questions

  1. What is the definition of a quadratic equation?

  2. What is completing the square used for?

  3. What is the x-intercept of a quadratic function?

  4. What is the axis of symmetry of a quadratic function?

  5. What is the purpose of solving quadratic equations?

  6. What is the vertex form of a quadratic function?

  7. What is the use of quadratic functions in real-world applications?

  8. What is the key point of completing the square?

  9. What is the importance of identifying the axis of symmetry in a quadratic function?

  10. What is the significance of solving quadratic inequalities?

  1. Discuss the importance of quadratic functions in real-world applications. (20 marks)

  2. Compare and contrast the methods of solving quadratic equations using factoring and completing the square. (20 marks)