Lecture Linear Function businness students

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1. Section 9B Linear Modeling

2. Linear Functions A linear function has a constant rate of change and a straight-line graph. Old example: The initial population of a town is 10,000 and increases by 500 people per year. Independent variable: Time (Year) Dependent variable: population Population is a function of Time (year). The constant rate of change is 500 people per year.

3. Year Population 10,000 5 12,500 10 15,000 15 17,500 20 20,000 40 30,000 Data Table old example: The initial population of town is 10,000 and increases by 500 people per year.

4. P = 10,000 + 500 x T Graph Equation old example: The initial population of town is 10,000 and increases by 500 people per year.

5. X Y 1 2 2 3 3 4 4 5 5 6 X Y 1 3 2 6 3 9 4 12 5 15 X Y 1 5 2 10 3 15 4 20 5 25 Case a Case B Case c

6.     We define rate of change/Slope of a linear function by: where (x 1 ,y 1 ) and (x 2 ,y 2 ) are any two ordered pairs of the function.  

7. A GRAPH FOR THE SLOPE

8. Year Population 10,000 5 12,500 10 15,000 15 17,500 20 20,000 40 30,000 Data Table old example: The initial population of town is 10,000 and increases by 500 people per year.

9. Year Population 10,000 5 12,500 10 15,000 15 17,500 20 20,000 40 30,000 old example: The initial population of town is 10,000 and increases by 500 people per year.         = 500 = 500 = 500 = 500 Rate of change is ALWAYS 500 (people per year).

10. Example: The table below gives the cost of US mail based on weight. Weight (independent) Postage cost (dependent) 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B Graph the cost as a function of weight Determine the slope from table Determine the slope from graph

11. determine the slope Weight (independent) Postage cost (dependent) 1 oz $0.37 2 oz $0.60 3 oz $0.83 4 oz $1.06 5 oz $1.29 6 oz $1.52 7 oz $1.75 9-B Rate of change is ALWAYS 0.23 (dollars per ounce).

12. Graph the cost as a function of weight

13. Graph the cost as a function of weight and determine the slope.

14. Graph the cost as a function of weight and determine the slope.

15. A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function Independent variable: price Dependent variable: demand (of pineapples) Demand is a function of price. ($2,80) and ($5,50)

16. ($2, 80 pineapples) and ($5, 50 pineapples) A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function For every dollar increase in price, the demand for pineapples decreases by 10.

17. ($2, 80 pineapples) and ($5, 50 pineapples). A linear function is used to describe how the demand for pineapples varies with the price. We know at a price of $2, the demand is 80 pineapples and at a price of $5, the demand is 50 pineapples. Find the rate of change (slope) for this function and then graph the function For every dollar increase in price, the demand for pineapples decreases by 10.

18. More Practice The water depth in a reservoir decreases at a rate of 0.25 inch per hour because of evaporation. How much does the water depth change in 6.5 hours? A tree increases its diameter by 0.2 inches per year by adding annual rings. How much does the diameter of the tree increase in 4.5 years?

19. More Practice A tree increases its diameter by 0.2 inches per year by adding annual rings. How much does the diameter of the tree increase in 4.5 years? Ans : The tree diameter increases by 0.2 inch per year. The rate of change is 0.2 inch per year. In 4.5 years, the tree diameter increases 0.9 inch.

20. More Practice The water depth in a reservoir decreases at a rate of 0.25 inch per hour because of evaporation. How much does the water depth change in 6.5 hours? Ans: The water depth decreases with respect to time at a rate of 0.25 inch per hours. The rate of change is —0.25 in./hr. In 6.5 hours, the water depth decreases 1.625 inches.

21. For linear functions: Slope = Rate of Change Positive Slope Negative Slope

22. The Rate of Change Rule Predict the change in demand for pineapples if the price increases by $3. change in demand = (-10 pineapples per dollar) x $3 = -30 pineapples change in dependent variable = (rate of change) x (change in independent variable) If the price of pineapples increases by $3, then the demand will decrease by 30 pineapples

23. dependent = initial value + ( rate of change x independent) or y = m x + b where m is slope and b is y intercept. General Equation for a Linear Function

24. y = m x + b

25. y = m x + b

26. Year Population 10,000 5 12,500 10 15,000 15 17,500 20 20,000 40 30,000 old example: The initial population of town is 10,000 and increases by 500 people per year.         = 500 = 500 = 500 = 500 Rate of change is ALWAYS 500 (people per year).

27. General Equation for a Linear Function Population: m = 500 and initial value = 10000 P = 10000 + 500T [ Y = 10000 + 500x ] Pineapple Demand: m = -10 and initial value = 100 D = 100 – 10p [ Y = 100 – 10x ]

28. Linear Functions Constant Rate of Change Straight Line Graph Dependent = Initial + Rate x Independent Y = mX + b