HPC2.2 – Linear

Functions & ModelsLearning Targets:

-Graph linear functions.-Draw and interpret scatter diagrams.

-Distinguish between linear and nonlinear relations.-Use your calculator to find the line of best fit.

-Construct a linear model using direct variation.

What is a linear function?

• A function of the form • Graph is a straight line• x is the independent variable; y = f(x) is the dependent variable.

( )f x mx b

DIRECT VARIATION

• Let x and y denote two quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that

y = kx.• k is called the constant of proportionality. • If y varies directly with x, then y is a linear function of x.

Ex 1) Suppose that a company has just purchased a new machine for its manufacturing facility for $120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. a) Write a linear function that expresses the book value of the

machine as a function of its age.

b) Graph the linear function.c) What is the book value of the machine after 4 years?

Ex 2)

a) Draw a scatter diagram (a.k.a. scatterplot).b) Select two points from the scatter diagram and find the equation of

the line containing the points selected.c) Graph the line found in part b on the scatter diagram.d) Use a graphing utility to find the line of best fit.e) Use a graphing utility to graph the line of best fit on the scatter

diagram.

x 3 5 7 9 11 13y 0 2 3 6 9 11

Ex 3) The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income. The following data represent the annual income, I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 30 years.

Annual Income, I ($) Loan Amount, L ($)

15,000 44,600

20,000 59,500

25,000 74,500

30,000 89,400

35,000 104,300

40,000 119,200

45,000 134,100

50,000 149,000

55,000 163,900

60,000 178,800

65,000 193,700

70,000 208,600

Ex 3) continued…a) Which is the independent variable? Dependent variable?b) Use a graphing utility to draw a scatter diagram of the data.c) Use a graphing utility to find the line of best fit to the data.d) Graph the line of best fit on the scatter diagram drawn in part b. e) Interpret the slope of the line of best fit.

f) Determine the loan amount that an individual would quality for if her income is $42,000. _________________

Ex 4) The monthly payment p on a mortgage varies directly with the amount borrowed B. If the monthly payment on a 15-year mortgage is $8.99 for every $1,000 borrowed, find a linear function that relates the monthly payment p to the amount borrowed B for a mortgage with the same terms. Then find the monthly payment when the amount borrowed is $175,000.