The Graph of a Function

Examples, solutions, and videos to help Algebra I students learn how to understand set builder notation for the graph of a real-valued function: {(x, f(x)) | x ∈ D} .
Students learn techniques for graphing functions and relate the domain of a function to its graph.

New York State Common Core Math Algebra I, Module 3, Lesson 11

Worksheets for Algebra I, Module 3, Lesson 11 (pdf)

Lesson 11 Summary

Graph of f: Given a function f whose domain D and the range are subsets of the real numbers, the graph of f is the set of ordered pairs in the Cartesian plane given by {(x, f(x)) | x ∈ D}

When we graph a function we want to think of “Input” and “Output”

Declare x an integer
Let f(x) = 2x + 1
Initialize G as {}
For all x from -1 to 4
Append (x, f(x)) to G
Next x
Plot G

Declare x a real
Let f(x) = 2x + 3
Initialize G as {}
For all x such that 2 ≤ x ≤ 8
Append (x, f(x)) to G
Next x
Plot G

Set Builder Notation
{(x, 2x + 1) | x is an integer, -1 ≤ x ≤ 4}
{(x, 2x + 3) | x is real, 2 ≤ x ≤ 8}

Exercise

4. Sketch the graph of the functions defined by the following formulas, and write the graph of f as a set using set- builder notation.
   (Hint: Assume the domain is all real numbers unless specified in the problem.)
   
   f(x) = 1/2x - 5
   f(x) = (x + 1)² - x², -3 ≤ x ≤ 3

Lesson 11 Exit Ticket

1. Perform the instructions for the following programming code as if you were a computer and your paper was the computer screen.
   Declare x an integer
   Let f(x) = 2x + 1
   Initialize G as {}
   For all x from -3 to 2
   Append (x, f(x)) to G
   Next x
   Plot G

2. Write three or four sentences describing in words how the thought code works.

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