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Graphing Functions: an interactive course

Published 8 Nov 20265 min read

Graphing Functions is an interactive graphing functions course that teaches you to read and sketch the main families of graphs. Its 31 short sections cover function notation, the coordinate plane, straight lines, transformations, quadratics, polynomials, reciprocal and rational functions with asymptotes, exponentials, square roots, and domain and range, nearly all with a slider explorer you can drag.

The aim is that you can look at an equation and picture its graph, or look at a graph and say what kind of equation made it.

What the course covers

Sections	Topic	What you do
1 to 3	What a function is, and f(x) notation	Use a function machine, then evaluate a function
4 to 5	The coordinate plane	Place points and read coordinates, then plot targets
6 to 8	Linear functions, slope and intercept	Drag m and b with rise and run arrows, then find a slope
9 to 11	Transformations: shifts, stretches and reflections	Transform a parabola against a fixed original, then name a transformation
12 to 14	Quadratic functions and parabolas	Slide a, b and c with the vertex and roots marked, then match an equation to a graph
15 to 17	Cubic and polynomial functions	Change the degree and see the turning points, then read a degree off a graph
18 to 20	Reciprocal and rational functions	Move a curve with its asymptotes, then find the asymptotes of a function
21 to 23	Exponential growth and decay	Compare bases, then tell linear, quadratic and exponential tables apart
24 to 25	Square root functions	Shift the curve, then match equations to graph shapes
26 to 28	Domain and range	Reveal them on a graph, then write them in interval notation
29 to 31	Reading graphs: intercepts, increasing and decreasing	Answer questions about a plotted curve

One family at a time

Each family gets its own short explanation, an explorer and a practice question. You learn what controls each shape: the slope and intercept of a line, the vertex and roots of a parabola, the degree of a polynomial, the asymptotes of a reciprocal curve and the base of an exponential.

Transformations

Transformations come early so every later family reuses them. f(x) + k moves a graph up, f(x − h) moves it right (the part that feels backwards), af(x) stretches it, and a minus sign reflects it. Once you know these, y = (x − 3)² + 1 is just y = x² moved right 3 and up 1.

Domain, range and reading graphs

The last sections pull it together: which inputs a function can take, which outputs it can give, where it crosses the axes, and where it rises and falls.

Try a question from the course

f(x) is the output when you put x in. For f(x) = 3x − 4, the output when x = 5 is f(5) = 3 × 5 − 4 = 11, so the graph passes through the point (5, 11).

Now evaluate a new function, then press Mark to check your answer in the course.

Press Mark to have your answer marked in the Graphing Functions course.

Who this course is for

Students in Algebra 1, Algebra 2 or precalculus, or GCSE and A-level Maths, who need to sketch and recognise graphs.
Students heading for calculus, which assumes you can picture graphs and read their features.
Science students who meet exponential decay, inverse relationships and quadratic motion in physics and chemistry.

What you need to know first

Substituting numbers into expressions, negative numbers, and simple equations. The Algebra course covers all three, and its quadratics sections help with the parabola part of this course.

How long it takes

There are 31 sections and most take a few minutes. Learners without a plan can open two sections of a course a day, in order, which takes 16 days. A plan removes the daily limit and lets you jump straight to any family of graphs, such as asymptotes or exponentials. Straight-line graphs also have their own guide, graphing linear equations.

Interactive calculatorGraphing CalculatorPlot any equation and find its roots, intercepts, turning points and gradients as you work through the course.

Guided courseCalculusThe next step: gradients of curves, areas under them, and what they tell you.

Common questions

Is this graphing functions course free?

You can take every section without paying, two sections a day, in order. A plan removes the daily limit and opens any section straight away.

Which types of function does the course cover?

Linear, quadratic, cubic and other polynomials, reciprocal and rational functions, exponential functions and square root functions, plus transformations of all of them.

What are parent functions?

The simplest member of each family, such as y = x, y = x², y = x³, y = 1/x, y = √x and y = 2ˣ. Every other graph in the family is a transformation of its parent, which is how this course teaches them.

How do you find the domain and range from a graph?

The domain is how far the graph stretches left and right, the x-values it uses. The range is how far it stretches up and down, the y-values it reaches. Section 27 shows both as strips on the graph.

What is an asymptote?

A line a graph gets closer and closer to but never reaches. y = 1/x has the x-axis and the y-axis as asymptotes.

Does the course cover trig or log graphs?

No. Sine and cosine graphs are in the Trigonometry course, and log graphs can be explored with the Log Graph Calculator.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

Solve it
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Graphing Functions
Graphing made easier: the patterns in numbers and functions used across STEM
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