1. CHAPTER 1: FUNCTIONS, GRAPHS, AND MODELS; LINEAR FUNCTIONS Section 1.4: Equations of Lines 1

2. SECTION 1.4: EQUATIONS OF LINES Slope-Intercept Form of the Equation of a Line y = mx + b where m is the slope, and b is the y-intercept Point-Slope Form of the Equation of a Line y – y 1 = m (x – x 1 ) where m is the slope, and (x 1, y 1 ) is a point on the line. notice that y and x are not ‘ spelled out ’ – that is because they are the variables that establish the linear relationship 2

3. SECTION 1.4: EQUATIONS OF LINES Write equations for each line, given the information provided. slope = 5, y-intercept at (-3, 0) slope =, passes through (-1, 5) slope = 0, passes through (-4, 2) slope is undefined, passes through (-4, 2) 3

4. SECTION 1.4: EQUATIONS OF LINES What happens when the slope = 0? We get a horizontal line in the form y = b. What happens when the slope is undefined? We get a vertical line in the form x = a. 4

5. SECTION 1.4: EQUATIONS OF LINES Another form … General Form of the Equation of a Line ax + by = c where a, b, and c are real numbers. the General Form is a way to express the line without any fractions. 5

6. SECTION 1.4: EQUATIONS OF LINES Find the equation of a line that passes through (-1, 5) and (2, 4). Express your answer in slope-intercept form and general form. 6

7. SECTION 1.4: EQUATIONS OF LINES The number of people (in millions) in US prisons or jails grew at a constant rate from 1990 to 2000, with 1.1.5 million people incarcerated in 1990 and 1.91 million incarcerated in 2000. Write an equation that models the number of prisoners, N, as a function of the year, x. The number of individuals incarcerated in 2005 is projected to be 2.29 million. Does your model agree? 7

8. SECTION 1.4: EQUATIONS OF LINES For a line, the slope measures the rate of change. However, not every equation is linear. What then? We can measure the Average Rate of Change The AROC is the slope of the line (called a secant line) connecting two points of interest on the curve. To find the AROC: 8

9. SECTION 1.4: EQUATIONS OF LINES The total Toyota hybrid vehicle units sold for the years between 1997 and 2001 can be approximated by the model S(x) = 2821x 3 – 75,653x 2 + 674,025x – 1,978,335 where x is the number of years after 1990. Find the average rate of change of total Toyota hybrid sales between 1997 and 1999. Interpret your answer. What is the relationship between the slope of the secant line joining the points (7,446) and (9,16,506) and your answer? 9

10. SECTION 1.4: EQUATIONS OF LINES What happens when real world data isn ’ t perfectly linear? We can use the AROC to determine a close fitting line. The enrollment (in thousands) in grades 9-12 of US schools for the years 1990-2002 is given in the table on the next slide. Create a scatterplot of the data. Find the AROC between 1990 and 2002. Write the equation of the line between 1990 and 2002. Graph your equation with the scatterplot. 10

11. SECTION 1.4: EQUATIONS OF LINES 11 Year, xEnrollment, y (in thousands) 199012,488 199112,703 199212,882 199313,093 199413,376 199513,697 199614,060 199714,272 199814,428 199914,623 200014,802 200115,058 200215,332

12. SECTION 1.4: EQUATIONS OF LINES Homework: pp. 63-69 1-29 every other odd, 31, 33, 35, 37, 41, 45, 51, 53, 55, 59, 65 12