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.