1. Investigating Properties of Linear Relations Cole buying a car.

2. Cole bought a new car for $27,500. This graph shows it’s value over the first three years. When will Cole’s car be worth $0.

3. Calculate the amount by which Cole’s car decreased in value between years 1 and 2. Value Year 2 – Value Year 1 = 17,500 – 22,500 = -5000 The value of the car decreased by approximately $5000.

4. Calculate the rate of change in the car’s value between years 1 and 3. Value Year 3 – Value Year 1 = 12,500 – 22,500 = -10000 The value of the car decreased by approximately $10000.

5. Slope (Note) A measure, often represented by m, of the steepness of a line The ratio comparing the vertical and horizontal distances (called the rise and run)

6. Calculate the slope of the graph between years 1 and 3. This means that the value of the car is decreasing by $5000 a year (approximately). How does the slope compare to your answer in part 1? It is the same!

7. Complete the first difference column in the table below. How does the first differences compare to the slope in part 3? Age of Car Year (x) Value of Car $ (y) First Difference $ (Δy) 027,500 122,675 217,850 313,025 48,200 22,675 – 27,500 = -4825 17,850 – 22,675 = -4825 13,025 – 17,850 = -4825 8,200 – 13,025 = -4825 They are approximately the same! A little different because we approximated from the graph.

8. Complete the following table. Why are the first differences different than in part 4? Age of Car Year (x) Value of Car $ (y) First Difference $ (Δy) 027,500 217,850 48,200 6-1450 17,850 – 27,500 = -9650 8,200 – 17,850 = -9650 -1,450 – 8,200 = -9650 Because the values of x (Age of the Car) are increasing by 2 instead of by 1.

9. Write an equation for the relation between the car’s value and it’s age. y=27,500 – 4,825x Y=-4,825x + 27,500 Slope First Differences Y-Intercept

10. Determine the x-intercept of the graph. Use it to tell when Cole’s car will be worth $0. How do you know? The x-intercept appears to be approximate 5.75 years. It is where the graph crosses the x-axis.

11. What is the connection between the first differences and the slope? The first differences equal the slope! BUT – only the difference in the x values are 1!!

12. When you calculated the slope, did it matter which points you chose? Explain. No, it does not! With a linear relationship, the slope is constant.

13. Use the graph to explain why the first differences were constant. A linear relationship has a constant slope … and therefore, constant first differences.