Unit 1: Advanced Algebra

Section A: Quadratic Functions and Their Applications

Welcome

Welcome to Section A: Quadratic Functions and Their Applications! In this section, you’ll dive into the world of quadratic functions, a foundational concept in algebra that has numerous real-world applications, from physics to finance.

Imagine

Imagine you’re an engineer calculating the trajectory of a projectile, or an economist modeling the cost and revenue curves of a product. Quadratic functions provide the tools to analyze and solve these kinds of problems.

Context

Previously, you’ve studied linear equations and their applications. Now, we’ll expand on that knowledge by introducing quadratic functions, which involve variables raised to the second power and lead to parabolic graphs.

Overview

This section covers exploring quadratic functions, graphing quadratic equations, solving them by factoring, completing the square, and using the quadratic formula. We’ll also delve into the applications of these functions in various fields.

Objectives

  • Understand the properties and behavior of quadratic functions.
  • Graph quadratic equations and analyze their key features, such as vertex and axis of symmetry.
  • Solve quadratic equations using factoring, completing the square, and the quadratic formula.
  • Apply quadratic functions to real-world problems in fields like physics, economics, and engineering.

Preparatory Guidance

Definitions and Pronunciations
  • Quadratic Function: A function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • Vertex: The highest or lowest point on the graph of a quadratic function.
  • Axis of Symmetry: A vertical line that divides the parabola into two mirror images, given by x = -\frac{b}{2a}.
Verbal Reading of Equations
  • For f(x) = ax^2 + bx + c, say “f of x equals a x squared plus b x plus c.”
  • For x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, say “x equals negative b plus or minus the square root of b squared minus four a c, all over two a.”
Problem-Solving Strategies
  • Explore the graph of a quadratic function by identifying its vertex, axis of symmetry, and intercepts.
  • Use the quadratic formula to find the roots of quadratic equations, applying it to various types of problems.
  • Factor quadratic expressions to solve equations and understand their structure.
  • Apply the completing the square technique to solve quadratic equations and understand vertex form.
  • Analyze real-world situations where quadratic functions are used, such as in projectile motion or profit optimization.

Considerations

How do quadratic functions model real-world phenomena? Why are quadratic equations important for solving various types of problems? In what ways can you apply quadratic functions in your daily life or future career?