Algebra - Graphing linear and quadratic functions

Linear Functions and Slope-Intercept Form: A linear function is represented by the equation $y = mx + c$, where the graph is a straight line. The value $m$ represents the gradient or slope; if $m > 0$, the line rises from left to right, and if $m < 0$, it falls. The constant $c$ is the y-intercept, the exact point $(0, c)$ where the line crosses the vertical y-axis.

Finding Intercepts: To find where a graph crosses the axes, set the opposite variable to zero. For the x-intercept (where the graph touches the horizontal axis), set $y = 0$ and solve for $x$. For the y-intercept (where the graph touches the vertical axis), set $x = 0$ and solve for $y$. This applies to both linear lines and quadratic curves.

Quadratic Functions and the Parabola: Quadratic functions follow the form $y = ax^2 + bx + c$ and create a symmetrical 'U' shaped curve called a parabola. If $a$ is positive, the parabola opens upwards (like a smile) and has a minimum point; if $a$ is negative, it opens downwards (like a frown) and has a maximum point.

The Vertex and Axis of Symmetry: The vertex is the 'turning point' of the parabola, representing its highest or lowest point. Every parabola is perfectly symmetrical across a vertical line called the Axis of Symmetry, which passes through the vertex. Visually, this line acts like a mirror, splitting the curve into two identical halves.

Roots and Zeros: The solutions to a quadratic equation $ax^2 + bx + c = 0$ are the x-intercepts of the graph, also known as roots or zeros. A parabola can cross the x-axis at two points (two real roots), touch it at exactly one point (one repeated root), or not cross it at all (no real roots), appearing to 'float' above or below the axis.

Gradient (Slope) Calculation: The gradient measures the 'steepness' of a line. It is visually understood as the 'rise over run'—the vertical change divided by the horizontal change between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.