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Linear Functions

Linear Functions. Lesson 1: Slope of a Line . 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

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Linear Functions

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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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