Algebra & Trig. 1 1.4 and 1.5 Linear Functions and Slope Slope is a measure of the steepness of a line and is denoted by the letter m. If a nonvertical line passes through two distinct points x, y 1 1 and x, y 2 2, then the slope of the line is given by the change in y rise y2 y1 m the change in x run x x. 2 1 Note: (i.e. A If a line rises from left to right, the slope is positive. If a line falls from left to right, the slope is negative. If a line is horizontal, the slope is zero. If a line is vertical, the slope of the line is undefined. vertical line has no slope.) The slope-intercept form of the equation of a nonvertical line is given by b is the y mx y -intercept, and b, where m is the slope of the line, x, y represents any point on the line. Notice that the slope-intercept form is solved for y. 1 P a g e
What is the slope and y-intercept of y = 3x + 8 What is the slope and y-intercept of 3y + 6x = 2 Find the slope of (0, 1) and (6, 8) To graph a linear equation in slope-intercept form... a) Identify the slope of the line and the y-intercept. b) Plot the y-intercept. c) Beginning at the point that you plotted, use the slope of rise the line to locate another point on the line. run The equation of a horizontal line is of the form y # (namely b ), since m 0. The equation of a vertical line is of the form x # (namely the x -intercept of the line). 2 P a g e
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals. Note that vertical lines are parallel to vertical lines and perpendicular to horizontal lines. Note that horizontal lines are parallel to horizontal lines and perpendicular to vertical lines. Slope and Parallel Lines 1. If two non-vertical lines are parallel, then they have the same slope 2. If two distinct non-vertical lines have the same slope then they are parallel 3. Two distinct vertical lines, both with undefined slopes are parallel Slope and Perpendicular Lines 1. If two non-vertical lines are perpendicular, then the product of their slopes is -1 (i.e. perpendicular lines have negative reciprocal slopes) 2. If the product of the slopes is -1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope 3 P a g e
Problem Type: You should be able to find the slope and the y- intercept of a line when given the equation of the line. (Remember to SOLVE FOR Y!) You should also be able to graph the line using the slope and the y-intercept. Graph the lines too. Ex.1: 3y 6 0 4 P a g e
x 7 Ex.2: 5 P a g e
Finding the Equation of a Line and Writing the Equation in Slope-Intercept Form You will be given at least one point that the line passes through as well as enough information to find the slope of the line (if it is not also given). You can then substitute this information into the point- slope form and finally solve for y in order to get the equation in slope-intercept form. Point Slope Form of a Linear Equation y y 1 = m(x x 1 ) where x 1 and y 1 are coordinates of the known point, m is the slope of the line and x and y are the variables of the equation To Solve: 1. Use point slope form of a linear equation y y 1 = m(x x 1 ) 2. Substitute known values for x, y and m 3. Rearrange the equation to be in standard form (ax +by = c) or slope intercept form (y = mx + b) Problem Type #1: Find the equation of the line in slopeintercept form. 2 7 m and the line passes through 0, 6 EX 1: 3. 6 P a g e
EX 2: The line passes through 1, 3) ( and ( 2, 2). EX 3: The line passes through ( 3, 2) and is perpendicular to the line 4x y 8. 7 P a g e
EX 4: The line passes through ( 6, 5) and is parallel to the line 6y 1. EX 5: The line is parallel to 7x 7y 4 and has the same y-intercept as 6x 3y 7. 8 P a g e
Number Living Alone (in Millions) Slopes as a Rate of Change Slope is defined as the ratio of the change in y to a corresponding change in x. It describes how fast y is changing with respect to x. For a linear function, slope may be interpreted as the rate of change of the dependent variable per unit change in the independent variable. Example Given the graph below. Find the slope for women and the slope for men and explain in words what the slopes mean. The graph is for the number of women and men living alone in the US. 18 (2005,17.2) 15 (1990,14) Women 12 Men (2005,12.7) 9 (1990,9) 6 3 1990 1995 2000 2005 Year 9 P a g e
For a non-linear function we can find the slope of the line by finding the slope of the secant line. The average rate of change between any two points is the slope of the line containing the two points, this line is called the secant line. The Average Rate of Change Let ( and ( de distinct points on the graph of a function f. The average rate of change of f from to, denoted by read delta y divided by delta x or :change in y divided by the change in x is Example Find the average rate of change of a) from b) 10 P a g e
Average Velocity Suppose that s function expresses an object s position, s(t), in terms of time, t. The average velocity of the object from to is Example The distance, s(t) in feet, traveled by a ball rolling down a ramp is given by the function where t is the time, in seconds, after the ball is released. Find the ball s average velocity from a) b) c) 11 P a g e