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Quadratic Functions

Examines quadratic equations, graphs, maximum/minimum points, and nature of roots.

📘 Topic Summary

Quadratic functions are a fundamental concept in mathematics that can be used to model various real-world phenomena, such as projectile motion and electrical circuits. This study guide will provide an overview of quadratic equations, their graphs, maximum/minimum points, and the nature of their roots.

📖 Glossary
  • Quadratic Equation: An equation of degree two in which the highest power of the variable is 2.
  • Vertex: The turning point or lowest point on a parabola's graph.
  • Axis of Symmetry: A line that passes through the vertex and divides the parabola into two equal parts.
  • Roots: The solutions to a quadratic equation.
  • Discriminant: A value calculated from the coefficients of a quadratic equation that determines the nature of its roots.
⭐ Key Points
  • Quadratic equations can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • The graph of a quadratic equation is a parabola that opens upwards or downwards.
  • The vertex of a parabola can be found using the formula (-b / (2a), f(-b / (2a)))
  • Quadratic equations can have one, two, or no real roots.
  • The discriminant (b^2 - 4ac) determines the nature of the roots: positive for two distinct real roots, zero for a repeated root, and negative for no real roots.
🔍 Subtopics
Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. For example, x^2 + 5x + 6 = 0 is a quadratic equation. The solutions or roots of a quadratic equation can be found by factoring the left-hand side, if possible, or using other methods such as completing the square.

Graphing Quadratic Functions

The graph of a quadratic function f(x) = ax^2 + bx + c is a parabola that opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola can be found using the formula (-b / 2a, f(-b/2a)). The x-coordinate of the vertex is the axis of symmetry.

Quadratic Formula

The quadratic formula is a general method for solving quadratic equations. It states that the solutions to the equation ax^2 + bx + c = 0 are given by the formula (-b ± √(b^2 - 4ac)) / 2a.

Nature of Roots

The nature of the roots of a quadratic equation depends on the value of the discriminant, b^2 - 4ac. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Solving Quadratic Equations

Quadratic equations can be solved using various methods such as factoring, completing the square, and the quadratic formula. The choice of method depends on the form of the equation and the desired solution.

Real-World Applications

Quadratic equations have many real-world applications in physics, engineering, economics, and computer science. For example, projectile motion, electrical circuits, and optimization problems can be modeled using quadratic equations.

Common Mistakes to Avoid

When solving quadratic equations, common mistakes to avoid include forgetting to check the solutions for extraneous roots, not simplifying the solutions properly, and misapplying the quadratic formula or other methods.

Practice Problems

Solve the following quadratic equations: x^2 + 4x - 3 = 0, x^2 - 7x + 12 = 0, and x^2 + 2x - 1 = 0.

🧠 Practice Questions

  1. What is the definition of a Quadratic Equation?

  2. What is the nature of the roots of a quadratic equation with a discriminant value of -16?

  3. What is the formula to find the vertex of a parabola?

  4. What is the axis of symmetry of a parabola?

  5. What is the nature of the roots of a quadratic equation with a discriminant value of 0?

  6. What is the quadratic formula for solving a quadratic equation?

  7. What is the maximum number of real roots that a quadratic equation can have?

  8. What is the term for the value calculated from the coefficients of a quadratic equation that determines the nature of its roots?

  9. What is the graph of a quadratic equation?

  1. Solve the quadratic equation x^2 + 4x - 3 = 0. (5 marks)

  2. Find the vertex of the parabola f(x) = x^2 + 2x - 1. (5 marks)

  3. Determine the nature of the roots of the quadratic equation x^2 - 7x + 12 = 0. (5 marks)

  4. Find the axis of symmetry of the parabola f(x) = x^2 + 3x + 2. (5 marks)

  5. Solve the quadratic equation x^2 + 2x - 3 = 0. (5 marks)

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

  2. Explain how to find the vertex of a parabola using the formula (-b / (2a), f(-b / (2a))). (20 marks)