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Understanding Slopes and Line Relationships

Learn how to find the slope of a line and identify parallel and perpendicular lines. Explore real-life applications and practice problems.

jamesgillis
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Understanding Slopes and Line Relationships

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  1. Warm Up Find the value of m. 1. 2. 3. 4.

  2. Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines.

  3. The slopeof a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

  4. AB Example 1A: Finding the Slope of a Line Use the slope formula to determine the slope of each line. Substitute (–2, 7) for (x1, y1) and (3, 7) for (x2, y2) in the slope formula and then simplify.

  5. AD Example 1C: Finding the Slope of a Line Use the slope formula to determine the slope of each line. Substitute (–2, 7) for (x1, y1) and (–2, 1) for (x2, y2) in the slope formula and then simplify. The slope is undefined.

  6. Remember! A fraction with zero in the denominator is undefined because it is impossible to divide by zero.

  7. One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

  8. Example 2: Transportation Application Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line.

  9. Example 2 Continued The slope is 65, which means Justin is traveling at an average of 65 miles per hour.

  10. Check It Out! Example 2 What if…? Use the graph below to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same.

  11. If a line has a slope of , then the slope of a perpendicular line is . The ratios and are called opposite reciprocals.

  12. Caution! Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear.

  13. UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3) Example 3A: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

  14. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) Example 3B: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

  15. CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3) Example 3C: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

  16. WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) Check It Out! Example 3a Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.

  17. Classwork/Homework • Pg. 185 (1-22 all)

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