The Graph of the Equation y = f(x)

Examples, solutions, and videos to help Algebra I students learn how to understand the meaning of the graph of y = f(x), namely {(x, y) | x ∈ D and y = f(x)}. Students understand the definitions of when a function is increasing or decreasing.

New York State Common Core Math Algebra I, Module 3, Lesson 12

Worksheets for Algebra 1

Lesson 12 Student Outcomes

• Students understand the meaning of the graph of y = f(x), namely {(x, y) | x ∈ D and y = f(x)}.
  
• Students understand the definitions of when a function is increasing or decreasing.

• Show Step-by-step Solutions

Lesson 12 Summary

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

When we write {(x, y) | y = f(x)} for the graph of y = f(x), it is understood that the domain is the largest set of real numbers for which the function f is defined.

The graph of f is the same as the graph of the equation y = f(x).

Increasing/Decreasing. Given a function f whose domain and range are subsets of the real numbers and is an interval contained within the domain, the function is called increasing on the interval I if
f(x₁) < f(x₂) whenever x₁ < x₂ in I

It is called decreasing on the interval if
f(x₁) > f(x₂) whenever x₁ > x₂ in I<

Example 1

In the previous lesson, we studied a simple type of instruction that computers perform called a for-next loop. Another simple type of instruction is an if-then statement.
Below is example code of a program that tests for and prints “True” when x + 2 = 4; otherwise it prints “False.”

Declare x integer
For all x from 1 to 4
If x + 2 = 4 then
Print True
else
Print False
End if
Next x

Example 2

Perform the instructions in the following programming code as if you were a computer and your paper was the computer screen:

Declare x integer
Initialize G as {}
For all x from 0 to 4
If x² - 4x + 5 = 2 then
Append x to G
else
Do NOT x append to G
End if
Next x
Print G

Lesson 12 Exit Ticket

1. Perform the instructions in the following programming code as if you were a computer and your paper was the computer screen:
2. Let f(x) = -1/2 x + 2, for in the domain 0 ≤ x ≤ 4.
   a. Write out in words the meaning of the set notation:
   {(x, y) | 0 ≤ x ≤ 4 and y = f(x)}
   b. Sketch the graph of y = f(x).

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