Exploring Linear Relationships

Exploring Linear Relationships

Understanding the connections between variables and how they change together in a straight line. Linear relationships are fundamental in data analysis and can provide valuable insights into patterns and trends. Let's dive into the world of linear relationships!

Linear Relationships

• A linear equation is in the form y = mx + b

• The slope, m, is the rate of change

  • The rate of change is constant between any two points

• The y-intercept, b, is the initial value

  • The initial value is the y-value when x = 0

Multiple Choice

What does the slope, m, represent in a linear equation?

• The rate of change

• The initial value

• The constant

• The y-value when x = 0

Slope: Rate of change

• The slope, m, in a linear equation represents the rate of change.

  • It determines how steep the line is and how much the dependent variable changes for each unit increase in the independent variable.

  • A steeper slope indicates a faster rate of change, while a flatter slope indicates a slower rate of change.

• The y-intercept, b, on the other hand, represents the initial value.

  • It is the starting point of the line when the independent variable is zero.

  • It is the y-value when x = 0.

Example 1

The total cost y of a phone plan for x number of months can be represented by the equation y = 75x + 100.

What does this slope (rate of change) represent?
What does this y-intercept (initial value) represent?

Example 1

The total cost y of a phone plan for x number of months can be represented by the equation y = 75x + 100.

Rate of change: $75 per month
Initial value: $100 fee when you sign up for the phone plan

Multiple Choice

The total cost y of a phone plan for x number of months can be represented by the equation y = 75x + 100.

Describe this relationship.

• Linear, but not proportional

• Both linear and proportional

• Neither linear nor proportional

• Proportional, but not linear

Example 2

A shoe store offers free points when you sign up for their rewards card. Then, for each pair of shoes purchased, you earn an additional number of points. The graph shows the total points earned for several pairs of shoes.
Find the rate of change. Then, interpret it.

Example 2

Rate of change:
30/2
=
15/1
=
15 points per pair of shoes purchased

Multiple Choice

How many points do you earn for signing up for the rewards card before you buy any shoes?

• 15 points

• 30 points

• 2 points

• 60 points

Example 3

The table shows how much money Ava has saved. Assume the relationship between the two quantities is linear.
What is the rate of change?
How can we find the initial value when it isn't included in the table?

Example 3

Rate of change: she saves $20 per month
Initial value: she started out with $50
Extend the table
Graph the line
Balance an equation

Example 4

Joan plans to add 12 photos to her photo album each week. After 8 weeks, there are 120 photos in the album. Assume the relationship is linear.
Write an equation in slope-intercept form to represent the relationship between photos, y, and weeks, x.

Multiple Choice

Joan plans to add 12 photos to her photo album each week. After 8 weeks, there are 120 photos in the album. Assume the relationship is linear.

What is the rate of change? (read carefully)

• 15 photos per week

• 12 weeks per photo

• 120 photos per 8 weeks

• 12 photos per week

Example 4

Joan plans to add 12 photos to her photo album each week. After 8 weeks, there are 120 photos in the album. Assume the relationship is linear.

y = 12x + 24