Section 9B Linear Modeling Pages 571-585

LINEAR constant rate of change 9-B Linear Modeling LINEAR constant rate of change

Understanding Rate of Change 9-B Understanding Rate of Change Example: The population of Straightown increases at a rate of 500 people per year. How much will the population grow in 2 years? 10 years? The population of Straightown varies with respect to time (year) with a rate of change of 500 people per year. P = f(y) In 2 years, the population will change by: (500 people/year ) x 2 years = 1000 people

Understanding Rate of Change 9-B Understanding Rate of Change Example/571: During a rainstorm, the rain depth reading in a rain gauge increases by 1 inch each hour. How much will the depth change in 30 minutes? The rain depth varies with respect to time (hour) with a rate of change of 1 inch per hour. D = f(h) In 30 minutes, the rain depth will change by: (1 inch/hour ) x (1/2 hour) = (1/2) inch

Understanding Rate of Change 9-B Understanding Rate of Change Example 27/583: The water depth in a lake decreases at a rate of 1.5 inches per day because of evaporation. How much does the water depth change in 6.5 days? in 12.5 days? The water depth varies with respect to time (days) with a rate of change of -1.5 inches per day. W = f(d) In 6.5 days, the water depth will change by: (-1.5 inches/day ) x (6.5 days) = -9.75 inches

Understanding Rate of Change 9-B Understanding Rate of Change Rate of Change Rule (p574): To calculate the change in dependent variable from the change in independent variable (change in dependent = (rate of x (change in independent variable) change) variable)

Understanding Linear Equations 9-B Understanding Linear Equations Example: The population of Straightown is 10,000 and increasing at a rate of 500 people per year. What will the population be in 2 years? The population of Straightown varies with respect to time (years) with an initial value of 10,000 and a rate of change of 500 people per year. P = f(y) P = 10000 + 500y P = 10000 + (500)(2) = 11000 people

Understanding Linear Equations 9-B Understanding Linear Equations Example: The rain depth at the beginning of a storm is ½ inch and is increasing at a rate of 1 inch per hour? What is the depth in the gauge after 3 hours? The rain depth varies with respect to time (hours) with an initial value of ½ inch and a rate of change of 1 inch per hour. D = f(h) P = 1/2+ (1)(h) P = 1/2 + (1)(3) = 7/2 inches or 3.5 inches

Understanding Linear Equations 9-B Understanding Linear Equations Example 27*/583: The water depth in a lake is 100 feet and decreases at a rate of 1.5 inches per day because of evaporation? What is the water depth after 6.5 days? The water depth varies with respect to time (days) with an initial value of 100 feet (1200 inches) and a rate of change of 1.5 inches per day. W = f(d) P = 1200-(1.5)(d) P = 1200-(1.5)(6.5) = 1200 – 9.75 = 1190.25 inches

Understanding Linear Equations 9-B Understanding Linear Equations General Equation for a Linear Function (p576): dependent var. = initial value + (rate of change x independent var.)

Graphing Linear Equations Example - Straightown: P = 10000 + 500y y P 10,000 1 10,500 2 11,000 3 11,500 5 12,500 10 15,000

Graphing Linear Equations Example – Rain Depth: D = 1/2 + (1)(h) h D 1/2 1 3/2 2 5/2 3 7/2 5 11/2 10 21/2

Graphing Linear Equations Example – Lake Water Depth: W = 1200 - (9.75)(d) h D 1200 1 1190.25 2 1180.5 3 1170.75 5 1151.25 10 1102.5

LINEAR constant rate of change (slope) 9-B Linear Modeling LINEAR constant rate of change (slope) straight line graph

Understanding Slope We define slope of a straight line by: where (x1,y1) and (x2,y2) are any two points on the graph of the straight line.

Understanding Slope Example: Calculate the slope of the Straightown graph.

Understanding Slope Example: Calculate the slope of the Water Lake Depth graph.

More Practice 33/583 The price of a particular model car is $15,000 today and rises with time at a constant rate of $1200 per year. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) How much will a new car cost in 2.5 years. 35/583 A snowplow has a maximum speed of 40 miles per hour on a dry highway. Its maximum speed decreases by 1.1 miles per hour for every inch of snow on the highway. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) At what snow depth will the plow be unable to move? 37/583 You can rent time on computers at the local copy center for $8 setup charge and an additional $1.50 for every 5 minutes. A) Clearly identify independent and dependent variable. B) Find a linear equation to describe the situation. C) How much time can you rent for $25?

Homework: Pages 582-583 # 23a-b(rate), 25a-b(rate), 28, 30, 34, 36, 38

More Practice 39/584 Suppose your dog weighed 2.5 pounds at birth and weighed 15 pounds after 1 year. A) Based on these two data points, find a linear function that describes how weight varies with age. B) Predict your dog’s weight at 5 and 10 years of age. C) Comment on the validity of the model. D) Sketch a graph of the weight function. 41/584 A Campus Republican fundraiser offers raffle tickets for $10 each. The prize for the raffle is a $350 television set, which must be purchased with proceeds from the ticket sales A) Find a function that gives the profit/loss for the raffle as it varies with the number of tickets sold. B) What is the profit/loss if 40 tickets are sold? C) How many tickets must be sold for the raffle sales to equal the cost of the prize? D) Sketch a graph of the profit/loss function.

43/584 A $1200 washing machine in a laundromat is depreciated for tax purposes at a rate of $75 per year. A) Find a function for the depreciated value of the washing machine as it varies with time. B) What is the depreciated value after 2 years? C) When will the depreciated value be $0. D) Sketch a graph of the function. 55/584 The cost of publishing a poster is $2000 for setting up printing equipment, plus $3 per poster printed. A) Find a function for the cost of publishing a poster as it varies with the number of posters printed. B) How much does it cost to print 2000 posters? C) How many posters can you print for $2800 D) Sketch a graph of the function.

Algebraic Linear Equations Slope Intercept Form y = b + mx b is the y intercept or initial value m is the slope or rate of change. More Practice/584: 45, 47, 49, 51

9-B Homework: Pages 584 # 40, 44, 48, 50, 54, 56, 58