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**Objectives**Linear Functions and Slopes • Find the slopes of lines • Write and graph linear equations • Model data with linear functions and make predictions**VOCABULARY**Linear equation Slope Point-slope form Slope-intercept form**(x2, y2)**y y2–y1 change in y (x1, y1) x2–x1 change in x x Slope Formula The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula (where The slope of a line is a number, m, which measures its steepness.**y2 – y1**5 – 3 2 1 m = = = = x2 – x1 2 4 – 2 y x Example: Find the slope of the line passing through the points (2,3) and (4,5). Use the slope formula. y2 y1 x1 x2 (4, 5) 2 (2, 3) 2**Your Turn**Find the slope of the line passing through each pair of points. (and (2, 8) (4, 5) and (8, 4)**Slope of Horizontal Lines**• Slope of a horizontal line is 0 • Equation of a horizontal line that passes through the point (a,b):**Slope of Vertical Lines**• Slope of a vertical line is undefined • Equation of a vertical line that passes through the point (a,b):******Because the slope of line is constant, it is possible to use**any point on a line and the slope of the line to write an equation of the line in point-slope form. All you need is a point and the slope.**Point-Slope Form**The Point-Slope form is derived from the slope formula. Slope Formula Change y2, x2 to just y and x. Multiple both sides by the denominator. Point-Slope Form**1**1 1 2 2 2 Example:The graph of the equation y – 3 = -(x – 4) is a line of slope m = - passing through the point (4,3). y m = - 8 (4, 3) 4 x 4 8 Point-Slope Form A linear equation written in the form y–y1 = m(x – x1) is in point-slope form. The graph of this equation is a line with slope mpassing through the point (x1, y1).**Write an equation in point-slope form for the line with**slope 4 that passes through the point . Then solve the equation for . Use the point-slope form of the equation. Substitute the given values Point-slope form Solve for Distribution property Combine like terms**y2 – y1**m = x2 – x1 Write an equation in point-slope form for the line passing through the points and . Then solve the equation for . Rule # 1: We need to find the slope, whether it is given to us or it needs to be calculated. Substitute the slope, , and either coordinate into the point-slope formula . To be continued **Point-slope form**Final answer The above answer is the slope-intercept form of the equation. where the slope and is the y-intercept.**Slope-Intercept Form**A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is b. To graph an equation in slope-intercept form: 1.Write the equation in the form y = mx + b. Identify m and b. 2.Plot the y-intercept (0,b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope.**y**x change in y 2 m = = 1 change in x (0,-4) (1, -2) Example: Graph the line y = 2x– 4. • The equation y = 2x– 4 is in the slope-intercept form. So, m = 2 and b = -4. 2. Plot the y-intercept, (0,-4). 3. The slope is 2. 2 4. Start at the point (0,4). Count 1 unit to the right and 2 units up to locate a second point on the line. 1 The point (1,-2) is also on the line. 5. Draw the line through (0,4) and (1,-2).**********General Form of the Equation of a Line**Every line has an equation that can be written in the general form where and are real numbers and and are not both zero.**Find the slope and the y-intercept of the line whose**equation is . We need to change the equation from general form to slope-intercept form. The slope is The y-intercept is**Using Intercepts to Graph**• Find the x-intercept. Let and solve for . Plot the point on the x-axis. • Find the y-intercept. Let and solve for . Plot the point on the y-axis. • Draw a line through the two points, using arrowheads on the ends to indicate the line continues in both directions indefinitely.**This is the graph of the equation .**y (0,4) (6,0) x 2 -2 Linear Equations The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.**Summary of Equations of Lines**• General form: • Vertical line: • Horizontal line: • Slope-intercept form: • Point-slope form:**Linear Model**Writing an equation of a line that models real data: If the data changes at a fairly constant rate, the rate of change is the slope. An initial condition would be the y-intercept. • Example:Suppose there is a flat rate of $.20 plus a charge of $.10/minute to make a phone call. Write an equation that gives the cost y for a call of x minutes. Note: The initial condition is the flat rate of $.20 and the rate of change is $.10/minute. Solution: y = .10x + .20**Linear Model**Writing an equation of a line that models real data: If the data changes at a fairly constant rate, the rate of change is the slope. An initial condition would be the y-intercept. • Example:The percentage of mothers of children under 1 year old who participated in the US labor force is shown in the table. Find an equation that models the data. Using (1980,38) and (1998,59)**Your Turn**The net sales for a car manufacturer were $14.61 billion in 2005 and $15.78 billion in 2006. Write a linear equation giving the net sales y in terms of x, where x is the number of years since 2000. Then use the equation to predict the net sales for 2007. Answer: y=1.17x+8.76, predicted sales for 2007 is $16.95 billion.