slope of a line formula with examples powerpoint presentation

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls for every unit it moves horizontally.

Formula

The slope of a line is calculated using the following formula:

Slope (m) = (Change in y) / (Change in x)

This can also be written as:

m = (y2 - y1) / (x2 - x1)

Where:

- (x1, y1) and (x2, y2) are any two points on the line.

Understanding the Formula

- Change in y: This represents the vertical change between the two points. It's the difference in the y-coordinates of the points.
- Change in x: This represents the horizontal change between the two points. It's the difference in the x-coordinates of the points.

Interpreting the Slope

- Positive Slope: A positive slope indicates that the line rises from left to right. The larger the slope, the steeper the line.
- Negative Slope: A negative slope indicates that the line falls from left to right. The larger the absolute value of the slope, the steeper the line.
- Zero Slope: A zero slope indicates a horizontal line.
- Undefined Slope: An undefined slope indicates a vertical line.

Examples

Example 1: Find the slope of the line passing through points (2, 3) and (5, 7).

- (x1, y1) = (2, 3)
- (x2, y2) = (5, 7)

m = (7 - 3) / (5 - 2) = 4 / 3

Therefore, the slope of the line is 4/3. This indicates a positive slope, meaning the line rises from left to right.

Example 2: Find the slope of the line passing through points (-1, 4) and (3, 4).

- (x1, y1) = (-1, 4)
- (x2, y2) = (3, 4)

m = (4 - 4) / (3 - (-1)) = 0 / 4 = 0

Therefore, the slope of the line is 0. This indicates a horizontal line.

Applications

Understanding the slope of a line is crucial in various fields, including:

- Mathematics: Solving equations, graphing lines, and analyzing geometric shapes.
- Physics: Calculating velocity, acceleration, and other physical quantities.
- Engineering: Designing structures, analyzing data, and optimizing processes.
- Economics: Modeling economic trends and forecasting future outcomes.

By understanding the concept of slope and its formula, we can gain valuable insights into the behavior of lines and use this knowledge to solve real-world problems.
Example 3: Finding the Slope of a Line from an Equation

Let's say we have the equation of a line in slope-intercept form:

y = 2x - 1

This equation is in the form y = mx + c, where:

- m is the slope
- c is the y-intercept (the point where the line crosses the y-axis)

In this case, we can directly identify the slope from the equation:

m = 2

Therefore, the slope of the line represented by the equation y = 2x - 1 is 2. This means the line rises 2 units for every 1 unit it moves to the right.

Here's how to visualize it:

1. Y-intercept: The equation tells us the y-intercept is -1. So, plot the point (0, -1) on the y-axis.
2. Slope: The slope is 2, which can be written as 2/1. This means for every 1 unit you move to the right, you move 2 units up.

1. What is a Line? A line is the set of points forming a straight path on a plane The slant (slope) between any two points on a line is always equal A line on the Cartesian plane can be described by a linear equation x-axis y-axis

2. Definition - Linear Equation Any equation that can be put into the form Ax + By  C = 0, where A, B, and C are Integers and A and B are not both 0, is called a linear equation in two variables. The graph will be a straight line. The form Ax + By  C = 0 is called standard form (Integer coefficients all on one side = 0)

3. Definition - Linear Equation The equation of a line describes all of the points on the line The equation is the rule for any ordered pair on the line 3x + 2y – 8 = 0 (4, -2) is on the line (5, 1) is not on the line x – 7y + 2 = 0 (4, -2) is not on the line (5, 1) is on the line Examples: Test the point by plugging the x and y into the equation

4. Slope Slope describes the direction of a line.

5. Guard against 0 in the denominator Slope If x 1  x 2 , the slope of the line through the distinct points P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) is: Why is this needed?

6. x-axis y-axis Find the slope between (-3, 6) and (5, 2) Rise Run -4 8 -1 2 = = (-3, 6) (5, 2)

7. Calculate the slope between (-3, 6) and (5, 2) x 1 y 1 x 2 y 2 We use the letter m to represent slope m

8. Find the Slopes (5, -2) (11, 2) (3, 9) Yellow Blue Red

9. Find the slope between (5, 4) and (5, 2). STOP This slope is undefined. x 1 y 1 x 2 y 2

10. x y Find the slope between (5, 4) and (5, 2). Rise Run -2 Undefined = =

11. Find the slope between (5, 4) and (-3, 4). This slope is zero. x 1 y 1 x 2 y 2

12. x y Rise Run -8 Zero = = Find the slope between (5, 4) and (-3, 4).

13. From these results we can see... The slope of a vertical line is undefined. The slope of a horizontal line is 0.

14. Find the slope of the line 4x - y = 8 Let x = 0 to find the y-intercept. Let y = 0 to find the x-intercept. (0, -8) (2, 0) First, find two points on the line x 1 y 1 x 2 y 2

15. Find the slope of the line 4x  y = 8 Here is an easier way Solve for y. When the equation is solved for y the coefficient of the x is the slope . We call this the slope-intercept form y = mx + b m is the slope and b is the y-intercept

16. x y Graph the line that goes through (1, -3) with (1,-3)

17. Sign of the Slope Which have a positive slope? Green Blue Which have a negative slope? Red Light Blue White Undefined Zero Slope

18. Slope of Parallel Lines Two lines with the same slope are parallel. Two parallel lines have the same slope.

19. Are the two lines parallel? L 1 : through (-2, 1) and (4, 5) and L 2 : through (3, 0) and (0, -2) This symbol means Parallel

20. x y Perpendicular Slopes 4 3 What can we say about the intersection of the two white lines?

21. Slopes of Perpendicular Lines If neither line is vertical then the slopes of perpendicular lines are negative reciprocals. Lines with slopes that are negative reciprocals are perpendicular. If the product of the slopes of two lines is -1 then the lines are perpendicular. Horizontal lines are perpendicular to vertical lines.

22. Write parallel, perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3). This symbol means Perpendicular

23. The Equation of a Line

24. Objectives Write the equation of a line, given its slope and a point on the line. Write the equation of a line, given two points on the line. Write the equation of a line given its slope and y-intercept.

25. Objectives Find the slope and the y-intercept of a line, given its equation. Write the equation of a line parallel or perpendicular to a given line through a given point.

26. Slope-intercept Form Objective Write the equation of a line, given its slope and a point on the line. y = mx + b m is the slope and b is the y-intercept

27. Write the equation of the line with slope m = 5 and y-int -3 Take the slope intercept form y = mx + b Replace in the m and the b y = mx + b y = 5 x + -3 y = 5x – 3 Simplify That’s all there is to it… for this easy question

28. Find the equation of the line through (-2, 7) with slope m = 3 Take the slope intercept form y = mx + b Replace in the y, m and x y = mx + b 7 = mx + b x y m 7 = 3 x + b 7 = 3 ( -2 ) + b 7 = -6 + b Solve for b 7 + 6 = b 13 = b Replace m and b back into slope intercept form y = 3x + 13

29. Write an equation of the line through (-1, 2) and (5, 7). First calculate the slope. Now plug into y, m and x into slope-intercept form . (use either x, y point) Solve for b Replace back into slope-intercept form Only replace the m and b

30. Horizontal and Vertical Lines If a is a constant, the vertical line though ( a, b ) has equation x = a . If b is a constant, the horizontal line though ( a , b, ) has equation y = b . (a, b)

31. Write the equation of the line through (8, -2); m = 0 Slope = 0 means the line is horizontal That’s all there is!

32. Find the slope and y-intercept of 2x – 5y = 1 Solve for y and then we will be able to read it from the answer. 5 5 5 Slope: y-int:

33. Write an equation for the line through (5, 7) parallel to 2x – 5y = 15.

34. We know the slope and we know a point. 7 = 2 + b 7 – 2 = b 5 = b Write an equation for the line through (5, 7) parallel to 2x – 5y = 15.

35. Write an equation for the line through (5, 7) parallel to 2x – 5y = 15.

36. The slope of the perpendicular. The slope of the perpendicular line is the negative reciprocal of m Flip it over and change the sign. Examples of slopes of perpendicular lines: -2 2.4 Note: The product of perpendicular slopes is -1 2 3 1 5 = -5 -2 1 12 5 -7 2

37. What about the special cases? What is the slope of the line perpendicular to a horizontal line? Well, the slope of a horizontal line is 0… So what’s the negative reciprocal of 0? 1 Anything over zero is undefined The slope of a line  to a horizontal line is undefined.

38. Write an equation in for the line through (-8, 3) perpendicular to 2x – 3y = 10. We know the perpendicular slope and we know a point. Isolate y to find the slope: 2x – 3y = 10 2x = 10 + 3y 2x – 10 = 3y 3 3 3 3 = 12 + b 3 – 12 = b -9 = b

39. Write an equation in standard form for the line through (-8, 3) perpendicular to 2x - 3y = 10.

40. Summary Slope-intercept form y is isolated Slope is m. y -intercept is (0, b)

41. Summary Vertical line Slope is undefined x-intercept is (a, 0) no y-intercept Horizontal line Slope is 0. y-intercept is (0, b) no x-intercept