Quadratic Function Basic - Algebra-1

1. Fundamental Concepts

Definition: A quadratic function is a polynomial function of degree 2 (n=2). Its general form is \(f(x) = ax^2 + bx + c\), where a, b, and c are constants, and \(a \neq 0\) (if \(a = 0\), the function becomes a linear function).
Parent Function: The simplest quadratic function, which is \(f(x)=x^2\) (in this case, \(a = 1\), \(b=0\), \(c = 0\)).

Form: Expressed as \(f(x)=ax^2 + bx + c\) (\(a\neq0\)), differing from linear functions with the form \(y=mx + b\) (n=1 polynomial).
Graphical Feature: All quadratic functions graph as parabolas (implied by the nature of degree 2 polynomials).
Symmetry: The parent function \(f(x)=x^2\) is symmetric about the y - axis (a property of quadratic functions, though specific axis of symmetry may vary for other quadratics).
Domain and Range (for parent function): The domain of \(f(x)=x^2\) is all real numbers, and its range is \([0,+\infty)\).

Quadratic functions:
- \(g(x)=5x^2 + 7x + 10\) (it has an \(x^2\) term with \(a = 5\neq0\), following the form \(ax^2+bx + c\)).
- \(f(x)=x^2\) (the parent function, with \(a = 1\neq0\), \(b = 0\), \(c=0\)).
Non - quadratic functions:
- Linear functions like \(f(x)=ax + b\) (they are degree 1 polynomials, lacking the \(x^2\) term).
- Functions where the highest power of x is not 2, such as constant functions (degree 0) or cubic functions (degree 3).

Evaluating the function: For a given input x, substitute it into the quadratic function to find the output. For example, for \(f(x)=x^2\), \(f(2)=2^2 = 4\), \(f(3)=3^2=9\).
Identifying coefficients: For a quadratic function in the form \(f(x)=ax^2+bx + c\), directly pick out a, b, and c. For \(f(x)=x^2\), \(a = 1\), \(b = 0\), \(c = 0\).
Analyzing basic properties: For the parent function \(f(x)=x^2\), determine features like the vertex (at \((0,0)\)), axis of symmetry (the y - axis, \(x = 0\)), and range (\([0,+\infty)\)) by examining its form and graph.