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Discovering Slopes: Understanding Line Steepness and Relationships

Learn how to find the slope of lines using formulas, identify parallel and perpendicular lines, and interpret slopes as rates of change. Practice with examples and determine line relationships graphically in a coordinate plane.

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Discovering Slopes: Understanding Line Steepness and Relationships

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  1. Warm Up Find the value of m. 1. 2. 3. 4. undefined 0

  2. Section 3-5, Slopes of lines

  3. Learning Targets I will determine the slope of a line. I will use the slopes of lines to identify parallel and perpendicular lines.

  4. Vocabulary rise run slope of a line zero slope undefined slope

  5. 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.

  6. 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. This line has ZERO slope, not no slope.

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

  8. 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.

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

  10. 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.

  11. If a line has a slope of , then the slope of a perpendicular line is . The ratios and are called opposite reciprocals. If you multiply and your product is -1.

  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 2A: 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. The products of the slopes is –1, so the lines are perpendicular.

  14. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) Example 2B: 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. The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

  15. CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3) Example 2C: 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. The lines have the same slope, so they are parallel.

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

  17. Check It Out! Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

  18. CRITICAL THINKING: If the slope of a line is greater than 0 but less than 1, what inequality can use use to find the slope (m) of a line perpendicular to the given line?

  19. Homework, Pg 186, #10 - 22

  20. 2.AB and XY for A(–2, 5), B(–3, 1), X(0, –2), and Y(1, 2) 3.MN and ST for M(0, –2), N(4, –4), S(4, 1), and T(1, –5) Lesson Quiz 1. Use the slope formula to determine the slope of the line that passes through M(3, 7) and N(–3, 1). m = 1 Graph each pair of lines. Use slopes to determine whether they are parallel, perpendicular, or neither. 4, 4; parallel

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