Graphs & Functions

The Coordinate Plane: Axes, Quadrants and Plotting Points

The coordinate plane is a grid made by two perpendicular number lines, the $x$-axis and the $y$-axis. Every point has an address $(x, y)$: how far across, then how far up.

Slope (Gradient) of a Line: Rise Over Run Explained

The slope, or gradient, of a line measures its steepness: how much $y$ changes for each step of $1$ in $x$. Between two points, $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$.

Equation of a Straight Line: y = mx + b and Point-Slope Form

Every non-vertical straight line can be written $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, the height where the line crosses the $y$-axis.

Systems of Equations: Solving by Substitution, Elimination and Graphing

A system of equations (or simultaneous equations) is a set of equations that must all be true at once. For two linear equations, the solution is the point where their lines intersect.

What Is a Function? Inputs, Outputs, f(x) and the Vertical Line Test

A function is a rule that gives exactly one output for each input. Writing $f(x)$ means "the output of the function $f$ when the input is $x$".

Domain and Range of a Function: How to Find Them

The domain of a function is the set of inputs it accepts. The range is the set of outputs it actually produces.

Composite and Inverse Functions: f(g(x)) and f⁻¹(x) Explained

A composite function $f(g(x))$ feeds the output of $g$ into $f$. An inverse function $f^{-1}$ undoes $f$: if $f(2) = 7$, then $f^{-1}(7) = 2$.

Transformations of Graphs: Shifts, Stretches and Reflections

A transformation moves or reshapes a graph. In $y = a\,f(x - h) + k$, $h$ shifts it right, $k$ shifts it up, $|a|$ stretches it vertically, and a negative $a$ reflects it in the $x$-axis.