Graphs of functions
A rule turns each input into one output. How can a single picture show every one of those pairs at once, and what can you read from it?
Where you are 5 inside
What this area is about
The graph of a function f is the set of all points (x, f(x)), one for each input x in its domain. So a point (a, b) lies on the graph exactly when f(a) = b. That one fact lets you go both ways: from a rule to a picture, and from a picture back to outputs, inputs and the story of what happened.
How these fit together
Making the picture. Graphing a function from a table works out some input and output pairs and plots each as a point.
Reading it. Reading values off a graph finds the output for a given input, and every input that gives a chosen output. Graphs that tell a story reads the shape in words, with no exact numbers, and draws a shape from a story.
Naming its landmarks. Features of a graph is an area of its own: intercepts, rising and falling intervals, turning points, symmetry, breaks, end behavior and rates of change.
Using two graphs. Solving equations with graphs and tables finds the inputs where two graphs meet, the solutions of f(x) = g(x), even when algebra cannot yet find them.
A path through it
Reading values comes before both the story graphs and the features of a graph. Plotting from a table and solving f(x) = g(x) with graphs fit anywhere.
Next, transformations move whole graphs by changing the rule, parent functions give the basic shapes to move, and piecewise functions draw different rules on different stretches.