Remember, there are four types of slope: Positive slope, negative slope, no slope and undefined slope. Positive Negative No slope Undefined slope
Definition of Slope Slope is defined as the ratio of the change in y to the change in x. We can represent this several ways: Where the capital Greek letter Delta (Δ) means “difference” or “change in”. The notation used in calculus for the first derivative -- the slope.
Slope Formula where “a” is the slope, when we put linear equations in slope intercept form, y = ax + b and “a” was the slope. This equation assumes we have two points, point one: (x1,y1), and point two: (x2,y2).
Finding the slope of a line from two points Given a graph with two points, we can calculate the slope of a line between them using the formula. It doesn’t matter which point we call point one or point two, but we MUST make sure we are consistent with which is which. Using point A as point 1 and point B as point 2: Using point B as point 1 and point A as point 2:
Example 1 Find the slope of the line that passes through (1,3) and (-2,-3). Then graph the line. First we’ll calculate slope using the formula: Then we’ll graph the line from the given points.
Slope Intercept Form of a Line The slope intercept form of the equation of a line with slope “a” and a y-intercept of b is: y = ax + b This form of a line is used to find the equation of a line if its slope and the y-intercept of the line are known. 6
Point-Slope Form of a Line The point-slope form of the equation of a line with slope m that passes through the point is: This form of a line is used to find the equation of a line if its slope and the coordinates of a point on the line are known. 7
Review What is the slope of a line containing the two points (2, 4) and (-3, -16) What is the equation of a line with the slope of -3 and passing through the point (2, 9)?
Parallel Lines Parallel lines are lines in the same plane that never intersect. Parallel lines have the same slope. -8 -6 -4 -2 2 4 6 8
Perpendicular Lines Perpendicular lines are lines that intersect to form a 900 angle. -8 -6 -4 -2 2 4 6 8 The product of the slopes of perpendicular lines is -1. (Slopes are negative reciprocals)
Example 1 Are these lines are parallel or perpendicular? y – 2 = 5x + 4 and -15x + 3y = 9 y = 5x + 6 3y = 9 + 15x y = 3 + 5x y = 5x + 3 The lines have the same slope. So they are parallel.
Example 2 Write an equation in slope-intercept form of a line containing the point (-3,-5) and parallel to the line y = 2x + 1. m = 2 First, we need the slope of the line y = 2x + 1. Second, we need to find out the slope of the line that is parallel to y = 2x + 1. Lastly, we use the point-slope formula to find our equation. y + 5 = 2x + 6 y = 2x + 1
Example 3 Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1. First, we need the slope of the line y = 2x + 1. m = 2 Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. Lastly, we use the point-slope formula to find our equation.
Example - Parallel Lines Are the two lines parallel? & The slope of the first line is - 4. The slope of the second line is also - 4. The lines are parallel. 14
Example - Perpendicular Lines Are the two lines perpendicular ? & The slope of the first line is The slope of the second line is The lines are perpendicular. 15