Quadratic Equations, Functions, Zeros, and Models

Lecture 3.2: Quadratic Equations, Functions, Zeros, and Models

Introduction and Reflection

• Location Snapshot: The lecture begins with a picture from Condom, France, featuring the presenter amidst a historical landscape, setting a reflective tone for the mathematical concepts to be explored. This lecture will delve into quadratic equations, functions, their zeros, and practical applications through various models.

1. Quadratic Functions

• Definition: A quadratic function is a polynomial function of degree two. It is generally expressed in the standard form: $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

• Graph of a Quadratic Function: The graph of a quadratic function is a parabola.

  • The parabola opens upwards if $a > 0$ (indicating a minimum value).

  • The parabola opens downwards if $a < 0$ (indicating a maximum value).

• Key Features of a Parabola:

  • Vertex: The highest or lowest point on the parabola. Its coordinates are $(h, k)$, where $h = -\frac{b}{2a}$ and $k = f(-\frac{b}{2a})$.

  • Axis of Symmetry: A vertical line passing through the vertex, given by the equation $x = -\frac{b}{2a}$. The parabola is symmetric with respect to this line.

  • y-intercept: The point where the parabola intersects the y-axis. This occurs when $x = 0$, so the y-intercept is $(0, c)$.

  • x-intercepts (Zeros/Roots): The points where the parabola intersects the x-axis. These are the values of $x$ for which $f(x) = 0$.

2. Quadratic Equations

• Definition: A quadratic equation is an equation that can be written in the standard form: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers with $a \neq 0$.

• Solving Quadratic Equations: Finding the values of $x$ that satisfy the equation.

  • Factoring: If the quadratic expression can be factored, set each factor to zero to find the solutions.

  • Square Root Property: Applicable for equations of the form $x^2 = k$, leading to $x = \pm \sqrt{k}$.

  • Completing the Square: A method to convert the quadratic equation into a perfect square trinomial, allowing the use of the square root property.

  • Quadratic Formula: The most general method for solving any quadratic equation. The solutions are given by:
    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Zeros of Quadratic Functions (Roots of Quadratic Equations)

• Definition: The zeros of a quadratic function are the values of $x$ for which $f(x) = 0$. These are equivalent to the roots or solutions of the corresponding quadratic equation $ax^2 + bx + c = 0$.

• The Discriminant ($\Delta$): The expression $b^2 - 4ac$ from the quadratic formula is called the discriminant. It determines the nature and number of the roots:

  • If $\Delta > 0$: There are two distinct real roots (two x-intercepts).

  • If $\Delta = 0$: There is exactly one real root (a repeated root, so one x-intercept, where the parabola touches the x-axis).

  • If $\Delta < 0$: There are two complex conjugate roots (no real x-intercepts; the parabola does not intersect the x-axis).

4. Quadratic Models and Applications

• Real-World Phenomena: Quadratic functions are used to model various real-world situations, often involving parabolic paths or optimization.

• Common Applications:

  • Projectile Motion: The height of an object thrown upwards often follows a parabolic path, modeled by a quadratic function ($h(t) = -16t^2 + v<em>0t + h</em>0$ for feet, or $h(t) = -4.9t^2 + v<em>0t + h</em>0$ for meters).

  • Optimization Problems: Finding maximum or minimum values in scenarios like maximizing profit, revenue, or minimizing costs, which correspond to the vertex of the parabola.

  • Area Problems: In geometry, maximizing the area of a shape given a fixed perimeter often leads to quadratic models.

  • Business and Economics: Revenue functions ($R(x) = ( ext{price per unit}) \times ( ext{number of units})$) and cost functions can often be modeled quadratically to find optimal production levels.