Functions - NoteRef

Quadratic Function Graph

Visualize functions on a graph.

Linear and Quadratic Functions: An Overview

Linear and quadratic functions are fundamental concepts in algebra, each with unique characteristics, behaviors, and visual representations.

A linear function is typically represented by the equation $f(x) = mx + b$ , where:

Ascending/Descending: The slope $m$ determines whether the line is ascending (positive $m$ ) or descending (negative $m$ ). If $m=0$ , the line is horizontal.
Angle: The angle or steepness of the line increases with the absolute value of $m$ ; larger values create a steeper line.
Crossing the x-axis: To find where the line crosses the x-axis (the root), set $f(x) = 0$ and solve for $x = -\frac{b}{m}$

Linear functions produce a straight line with a constant slope, indicating a steady rate of change.

A quadratic function is generally represented by the equation $f(x) = ax^2 + bx + c$ , where:

$a$ , $b$ , and $c$ are constants, with $a \neq 0$ (if $a=0$ , the function is linear).
The function forms a parabola, a symmetrical, curved shape.

Direction of the Parabola: The coefficient $a$ affects the direction the parabola opens. If $a > 0$ , the parabola opens upwards (concave up); if $a < 0$ , it opens downwards (concave down).
Vertex: The vertex of the parabola is its maximum or minimum point, depending on the direction of opening. The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$ , and substituting this back into the function gives the y-coordinate, yielding the vertex $(x, y)$ .
Axis of Symmetry: The line x=−b2ax = -\frac{b}{2a}x=−2ab is the axis of symmetry, dividing the parabola into two mirror-image halves.
Roots (x-intercepts): To find where the parabola crosses the x-axis, set $f(x) = 0$ and solve for $x$ using the Bhaskara formula (also known as the quadratic formula):
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$