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3.3 Slopes

3.3 Slopes. The Slope of a Line. An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.”.

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3.3 Slopes

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  1. 3.3 Slopes

  2. The Slope of a Line An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.” One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line. Slide 3.3-3

  3. Objective 1 Find the slope of a line, given two points. Slide 3.3-4

  4. The slope of a line is the same for any two points on the line. Find the slope of a line, given two points. (cont’d) The slope of a line can be found through two nonspecific points. This notation is called subscript notation, read x1 as “x-sub-one” and x2as “x-sub-two”. Moving along the line from the point (x1,y1) to the point (x2,y2), we see that y changes byy2− y1units. This is the vertical change (rise). Similarly, xchanges by x2−x1 units, which is the horizontal change (run). The slope of the line is the ratio of y2−y1 to x2−x1. Slope Formula The slope m of a line through the points (x1,y1) and (x2,y2) is Slide 3.3-7

  5. Find the slope of a line, given two points. To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y-values, which is the difference 6 − 1 = 5 units. The horizontal change, orrun, is the change in the x-values, which is the difference 5 − 2 = 3 units. The slope is the ratio of the vertical change in yto the horizontal change in x. Slide 3.3-5

  6. CLASSROOM EXAMPLE 1 Finding the Slope of a Line Find the slope of the line. Solution: Slide 3.3-6

  7. and yield the same slope. Make sure to start with the x- and y-values of thesame point and subtract the x- and y-values of the other point. CLASSROOM EXAMPLE 2 Finding Slopes of Lines Find the slope of the line through (6, − 8) and (−2, 4). Solution: Slide 3.3-8

  8. Find the slope of a line, given two points. (cont’d) Orientation of Lines with Positive and Negative Slopes A line with a positiveslope rises (slants up) from left to right. A line with a negative slope falls (slants down) from left to right. Slopes of Horizontal and Vertical Lines Horizontal lines, with equations of the form y= k, have slope 0. Vertical lines, with equations of the form x= k, have undefined slopes. Slide 3.3-9

  9. CLASSROOM EXAMPLE 3 Finding the Slope of a Horizontal Line Find the slope of the line through (2, 5) and (−1, 5). Solution: Slide 3.3-10

  10. CLASSROOM EXAMPLE 4 Finding the Slope of a Vertical Line Find the slope of the line through (3, 1) and (3,−4). Solution: undefined slope Slide 3.3-11

  11. Objective 2 Find the slope from the equation of a line. Slide 3.3-12

  12. Find the slope from the equation of a line. (cont’d) The slope, −3 is found, which is the same number as the coefficient of x in the given equationy = −3x + 5. It can be shown that this always happens, as long as the equation is solved for y. Finding the Slope of a Line from Its Equation Step 1:Solve the equation fory. Step 2:The slope is given by the coefficient ofx. Slide 3.3-14

  13. CLASSROOM EXAMPLE 5 Finding Slopes from Equations Find the slope of the line 3x + 2y = 9. Solution: Slide 3.3-15

  14. Objective 3 Use slopes to determine whether two lines are parallel, perpendicular, or neither. Slide 3.3-16

  15. Lines are perpendicularif they intersect at a 90° angle. The product of the slopes of two perpendicular lines, neither of which is vertical, is always − 1. This means that the slopes of perpendicular lines are negative (or opposite) reciprocals — if one slope is the nonzero number a, the other is . The table to the right shows several examples. Use slopes to determine whether two lines are parallel, perpendicular, or neither. Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes. Slide 3.3-17

  16. Use slopes to determine whether two lines are parallel, perpendicular, or neither. (cont’d) Slopes of Parallel and Perpendicular Lines Two lines with the same slope are parallel. Two lines whose slopes have a product of − 1 are perpendicular. Slide 3.3-18

  17. CLASSROOM EXAMPLE 6 Deciding Whether Two Lines Are Parallel or Perpendicular Decide whether the pair of lines is parallel, perpendicular, or neither. Solution: The product of their slopes is − 1, so they are perpendicular Slide 3.3-19

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