1
Read and write a line four ways
Move between slope-intercept, point-slope, standard form and two given points, and pick the form that makes the question easy.
Course · Algebra
Lines, Functions and Graphs
Put algebra on a picture. Slope and every form of a line, then the definition that organises the rest of the path — a function — with its notation, its domain, the transformations that move its graph, and its inverse.
m = (y₂ − y₁)/(x₂ − x₁)
y = mx + b y − y₁ = m(x − x₁)
m₁m₂ = −1 perpendicular
(f ∘ g)(x) = f(g(x)) f⁻¹(f(x)) = x
What this course is about
- Subject
- Algebra
- Lessons
- 14
- Format
- Interactive
Overview
What you will be able to do
By the end you can turn a linear equation or function rule into a checked graph, value or related rule, and justify each decision from the definition rather than from the picture alone.
2
Apply the definition of a function
Decide whether a rule, a table or a graph defines a function, and say which input breaks it when one does.
3
State a domain and a range
Find what a formula excludes — a zero denominator or a negative under a square root — write the domain in interval notation, and justify the range of a line, square or absolute-value rule from its outputs.
4
Transform, compose and invert
Map a known point under a·f(b(x − h)) + k, compose two functions in the right order with the correct domain, and find and verify the inverse of a line or a restricted square.
Lessons
14 lessons
Lines and the half-planes they bound come first. “What a Function Is” defines a function; the rest of the course develops its notation, domain and range, piecewise rules, graph transformations, composition and inverse.
The Coordinate PlaneOrdered pairs, axes and quadrants: where a graph lives.Not marked
Graphing a Linear EquationFrom a table of solutions to a straight line, and the intercept shortcut.Not marked
SlopeOne number for steepness and direction, and why it does not depend on the two points you pick.Not marked
Slope-Intercept Formy = mx + b, and reading a line straight off its equation.Not marked
Point-Slope and Standard FormBuilding a line from a point and a slope, and the form that treats x and y alike.Not marked
Parallel and Perpendicular LinesTwo slope tests, and the vertical case that neither test covers.Not marked
Linear Inequalities in Two VariablesA boundary line, and which side of it is the answer.Not marked
What a Function IsExactly one output for each input, and the test that checks it.Not marked
Function Notationf(x) is a value, not a product.Not marked
Domain and RangeWhat goes in, what comes out, and what is excluded.Not marked
Piecewise FunctionsOne function, several formulas, and the joins between them.Not marked
Transformations of GraphsShift, stretch and reflect by mapping points from a known graph.Not marked
Composition of FunctionsFeeding one function into another, in a fixed order.Not marked
Inverse FunctionsUndoing a function, and the restriction that is usually needed.Not marked
Recommended background
Solving and rearranging linear equations.
Practice suggestions
After each complete example, cover its answer and finish the faded rehearsal. The supplied first decision is guidance; the remaining algebra, graph check and explanation are yours.
Sketch or predict before you reveal a lab result. A graph is most useful when it tests a claim you already made, not when it makes the claim for you.
For transformations and composition, trace one input or one known point all the way through. The order becomes visible in the intermediate value, and a wrong order cannot hide behind a plausible final formula.
What it does not cover
Continuity and limits. The word "smooth" is used informally here; making it precise is calculus.
Polynomial and rational graphs beyond lines, which need factoring and arrive in Polynomials and Factoring and Rational and Radical Expressions.
Conic sections. Circles and ellipses are not functions of x, and this course is about functions.