Representing Linear Non-proportional Relationships Essential Question? How can you use tables, graphs, mapping diagrams, and equations to represent linear non-proportional situations? 8.F.3

Common Core Standard: 8.F.3 ─ Define, evaluate, and compare functions. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Objectives: Understand that a linear function has a constant rate of change called slope. Identify whether a relationship is a linear function from a diagram, table of values, graph, or equation.

Curriculum Vocabulary Linear Equation (ecuación lineal): An equation whose solutions form a straight line on a coordinate plane. Slope Intercept Form (forma de pendiente-intersección): A linear equation written in the form y = mx+b, where m represents the slope and b represents the y-intercept. y-intercept (intersección con el eje y): The y-coordinate of the point where the graph of a function crosses the y-axis.

LINEAR RELATIONSHIPS Let’s examine the following situation: Hertz rental car charges $35 per day plus $0.50 per mile to rent one of their pick-up trucks. Create a table, write an equation, and draw a graph for this situation. Number of miles driven (x) Cost in dollars (y) What does it cost to drive 0 miles (initial value)? What other numbers would you choose for the table?

LINEAR RELATIONSHIPS Hertz rental car charges $35 per day plus $0.50 per mile to rent one of their pick-up trucks. Create a table, write an equation, and draw a graph for this situation. Number of miles driven (x) 5 10 15 20 Cost in dollars (y) 35 37.50 40 42.50 45 Do I have a Function? Do I have a Constant Rate of Change? What is my Constant Rate of Change? What is my starting point (initial value)? YES! 𝟓 𝟏𝟎 = 𝟏 𝟐 =𝟎.𝟓𝟎 (0,35)

LINEAR RELATIONSHIPS

LINEAR RELATIONSHIPS A LINEAR RELATIONSHIP can be described by an equation in the form 𝒚=𝒎𝒙+𝒃, where 𝒎 is the SLOPE and b is the y-INTERCEPT. The y-intercept is the same as the initial value and should always be written as the ordered pair (0,b) even though many books don’t show it that way.

LINEAR RELATIONSHIPS Let’s continue with the Hertz rental car. Number of miles driven (x) 5 10 15 20 Cost in dollars (y) 35 37.50 40 42.50 45 𝒎= 𝟓 𝟏𝟎 = 𝟏 𝟐 =𝟎.𝟓 (0,35) 𝒃=𝟑𝟓 𝒚= 𝟏 𝟐 𝒙+𝟑𝟓 or 𝒚=𝟎.𝟓𝒙+𝟑𝟓 What is the slope? What is the y-intercept? What is the linear equation?

Number of miles driven (x) LINEAR RELATIONSHIPS Now let’s graph the function. What would be the best scale for my graph? Number of miles driven (x) Cost in dollars (y) 35 5 37.5 10 40 15 42.5 20 45

Number of miles driven (x) LINEAR RELATIONSHIPS We have now represented the linear relationship as a table, an equation, and a graph. Number of miles driven (x) Cost in dollars (y) 35 5 37.5 10 40 15 42.5 20 45 50 45 40 35 30 25 20 15 10 5 Cost in dollars 𝑦= 1 2 𝑥+35 5 10 15 20 Number of miles driven

What other numbers would you choose for the table? LINEAR RELATIONSHIPS Now let’s consider the following situation: José went to the Riverside County Fair & Date Palm Festival in Indio. Cost of admission was $20. He also decided to go on some rides. Ride tickets cost $2 each. Create a table, equation, and graph to represent this scenario. Number of tickets (x) Price paid in dollars (y) What does it cost José to enter the festival, even if he doesn’t purchase any ride tickets (initial value )? What other numbers would you choose for the table?

LINEAR RELATIONSHIPS José went to the Riverside County Fair & Date Palm Festival in Indio. Cost of admission was $20. He also decided to go on some rides. Ride tickets cost $2 each. Create a table, equation, and graph to represent this scenario. Number of tickets (x) 1 2 3 4 Price paid in dollars (y) 20 22 24 26 28 Do I have a Function? Do I have a Constant Rate of Change? What is my Constant Rate of Change? What is my starting point (initial value)? YES! 𝟐 𝟏 =𝟐 (0,20)

LINEAR RELATIONSHIPS Let’s continue: What is the slope? (0,20) 𝒃=𝟐𝟎 Number of tickets (x) 1 2 3 4 Price paid in dollars (y) 20 22 24 26 28 𝒎= 𝟐 𝟏 =𝟐 (0,20) 𝒃=𝟐𝟎 𝒚=𝟐𝒙+𝟐𝟎 What is the slope? What is the y-intercept? What is the linear equation?

Price paid in dollars (y) LINEAR RELATIONSHIPS Now let’s graph the function. What would be the best scale for my graph? Number of tickets (x) Price paid in dollars (y) 20 1 22 2 24 3 26 4 28 Would it make sense to connect these points with a line? Why / Why not?

Price paid in dollars (y) LINEAR RELATIONSHIPS We have now represented the linear relationship as a table, an equation, and a graph. Number of tickets (x) Price paid in dollars (y) 20 1 22 2 24 3 26 4 28 50 45 40 35 30 25 20 15 10 5 Price paid in dollars 𝑦=2𝑥+20 1 2 3 4 Number of tickets purchased

Try This One! Your mom fills the 12 gallon gas tank in her car with gas. On average her car gets 37 miles to the gallon. Create a table, equation, and graph showing how far your mom can drive before she runs out of gas.