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Understanding the Slope of a Line: Key Concepts and Applications

In this lesson, we explore the concept of slope in linear functions, focusing on its measurement through rise and run. Learn to calculate slope using coordinates, identify whether a line has a positive or negative slope, and understand the significance of horizontal and vertical lines. We will demonstrate how to determine slopes from given points and apply this knowledge to find unknown coordinates. Engage with practical examples and exercises to solidify your understanding of line steepness and its mathematical implications.

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Understanding the Slope of a Line: Key Concepts and Applications

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  1. Linear Functions Lesson 1: Slope of a Line

  2. Today’s Objectives • Demonstrate an understanding of slope with respect to: rise and run; rate of change; and line segments and lines, including: • Determine the slope of a line or line segment using rise and run • Classify a line as having either positive or negative slope • Explain the slope of a horizontal or vertical line • Explain why the slope can be found using any two points on the graph of the line or line • Draw a line segment given its slope and a point on the line

  3. Vocabulary • Slope • The measure of a lines steepness • (vertical change/horizontal change) • Rise • The vertical change of a line • Run • The horizontal change of a line

  4. Slope of a Line • The slope of a line segment is a measure of its steepness • This means a comparison between the vertical change and the horizontal change: • The vertical change (is called the rise • The horizontal change is called the run • Slope is normally represented by the lowercase m. • We can calculate the slope in several ways such as by counting or using coordinates of two points on the line • m = slope = =

  5. Counting Slope formula Slope of a Line Slope = rise/run Slope = -3/6 Slope = -1/2 Slope = rise/run = y2-y1/x2-x1 Slope = [-2-1]/[4-(-2)] Slope = -3/6 = -1/2 (x1,y1) A (-2,1) Down 3 (x2,y2) B(4,-2) Right 6

  6. Slope of a Line • If the line segment goes downward from left to right, it will have a negative slope. (rise = negative) • If the line segment goes upwards from left to right, it will have a positive slope. (rise = positive) • *The steeper the line goes up or down, the greater the slope.

  7. Horizontal and Vertical Lines • If a line is horizontal, that is, the rise is equal to zero, then the slope will also be equal to zero. • Slope == = 0 • If a line is vertical, that is, the run is equal to zero, then the slope of the line will be undefined. • Slope = == = = undefined

  8. Example 1) You do • Find the slopes of the following line segments. Which line segment has the steepest (greatest) slope? Graph the line segments. • A) A(-1, 7) B(4, -3) • B) A(-20, 3) B(-4, -5)

  9. Solutions (-1,7) (-20,3) (4,-3) (-4,-5) Slope of line a) = -10/5 = -2 Slope of line b) = -8/16 = -1/2 Line segment in a) is steeper than line segment b)

  10. Finding Unknown Coordinates • We can also use the slope formula to find the coordinates of an unknown point on the line when we know the slope and another point on the line. • Example 2) Given a line that passes through R(5,-6) and has a slope of -2/7, determine another point, T, that the line passes through. • Solution: • We can set one unknown coordinate to equal zero, then solve for the final remaining unknown coordinate. For example: • Let x = 0, solve for y • = = ; y + 6 = • y = 10/7 – 6 = -32/7. So a second point, T, on the graph could be (0, -32/7)

  11. Finding Unknown Coordinates • Another way to find a second point is to simply count out the rise and the run from the one known point. • In this case we can read the slope as -2/7 or 2/-7, so we could find two possible points:(-2, -4) or (12, -8) (-2, -4) Known point (5, -6) (12, -8)

  12. Wall Quiz!

  13. Homework • Pg. 339-343 • #4,6,9,10,16,17,20,22,24,26,29

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