N. Linear Functions

N.Exploring Linear Functions

N. 1 Identify linear functions from graphs and equations
N. 2 Identify linear functions from tables
N. 3 Complete a table and graph a linear function

Identifying Linear Functions

A function whose graph is a straight line is a linear function. It can be identified from graphs, equations, or tables. The graph of a nonlinear functions is not a straight line. Linear functions have a constant rate of change and can be represented by equations in the form y = mx + b, where m is the slope and b is the y-intercept. Linear functions are widely used in various fields, including mathematics, physics, and economics.

Multiple Choice

These are linear functions

• TRUE

• FALSE

Linear Function:

y = mx + b is the equation form of a linear function. It represents a straight line on a graph, where m is the slope and b is the y-intercept. Linear functions are used to model relationships between two variables that have a constant rate of change.

Multiple Choice

What is the equation form of a linear function?

• y = mx + b

• y = ax^2 + bx + c

• y = e^x

• y = ln(x)

Linear vs Nonlinear Functions

Linear functions have a constant rate of change, while nonlinear functions do not. To determine if a function is linear, look for a constant first difference in the range. If the first difference is constant, the function is linear. If not, it is nonlinear. Use tables to identify linear functions by checking for a constant rate of change in the domain.

Multiple Choice

These are linear functions

• FALSE

• TRUE

Constant Rate of Change

Trivia: One way to determine if a function is linear is by checking for a constant rate of change in the domain. This means that for every unit increase in the independent variable, there is a consistent change in the dependent variable. Nonlinear functions do not exhibit this property.

• Linear functions have a constant slope.
• Nonlinear functions have a varying slope.

Multiple Choice

What is one way to determine if a function is linear?

• Look for a constant first difference in the range

• Check for a constant rate of change in the domain

• Examine the title of the passage

• Use tables to identify nonlinear functions

Multiple Choice

This slope in constant so we have a linear function

• Sometimes

• No

• Yes

• I don't know

Exploring Linear Functions

Linear functions are mathematical functions that can be represented by a straight line. They have a constant rate of change, also known as the slope. To determine if a function is linear, check if the rate of change in the domain is constant. Use the formula (Y₂ - Y₁) / (X₂ - X₁) to calculate the slope between any two points. Complete a table and graph the function to visualize the linear relationship. Remember, the general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept.

Reorder

Reorder the following from top to bottom

• 0

• 3

• 6

1

2

3

• 1
• 2
• 3

Slope Formula:

(Y₂ - Y₁) / (X₂ - X₁) is the formula to calculate the slope of a linear function. It represents the change in the y-coordinates divided by the change in the x-coordinates. The slope determines the steepness of the line and can be positive, negative, or zero. Remember, the numerator represents the vertical change and the denominator represents the horizontal change.

Multiple Choice

What is the formula to calculate the slope of a linear function?

• (Y₂ - Y₁) / (X₂ - X₁)

• (X₂ - X₁) / (Y₂ - Y₁)

• (Y₁ - Y₂) / (X₂ - X₁)

• (X₁ - X₂) / (Y₁ - Y₂)

Poll

From left to right, what are the correct solutions?

• The solution should be 9, 11, -7

• The solution should be -7, 11, 9

• The solution should be 11, -7, 9

• The solution should be -7, 9, 11

Open Ended

REVIEW:

What is the slope?

What is the y-intercept?

What does the slope mean? Is it positive or negative?

What does the y-intercept mean?

What is f(9) equal to?

Multiple Select

Which concepts did we explore in today's notes?

• Graphing linear functions

• Completing tables

• Identifying linear functions

• Understanding slope and y-intercept