The Slope of a Line. 1. Find the slope of a line through two given points 2. Find the slope of a line from its graph

Transcription

1 6.4 The Slope of a Line 6.4 OBJECTIVES 1. Find the slope of a line through two given points 2. Find the slope of a line from its graph We saw in Section 6.3 that the graph of an equation such as 2 3 is a straight line. In this section we want to develop an important idea related to the equation of a line and its graph, called the slope of a line. Finding the slope of a line gives us a numerical measure of the steepness or inclination of that line. NOTE Recall that an equation such as 2 3 is a linear equation in two variables. Its graph is alwas a straight line. Q( 2, 2 ) P( 1, 1 ) Change in 2 1 Change in 2 1 ( 2, 1 ) NOTE 1 is read sub 1, 2 is read sub 2, and so on. The 1 in 1 and the 2 in 2 are called subscripts. To find the slope of a line, we first let P( 1, 1 ) and Q( 2, 2 ) be an two distinct points on that line. The horizontal change (or the change in ) between the points is 2 1. The vertical change (or the change in ) between the points is 2 1. We call the ratio of the vertical change, 2 1, to the horizontal change, 2 1, the slope of the line as we move along the line from P to Q. That ratio is usuall denoted b the letter m, and so we have the following formula: Definitions: The Slope of a Line NOTE The difference 2 1 is sometimes called the run between points P and Q. The difference 2 1 is called the rise. So the slope ma be thought of as rise over run. If P( 1, 1 ) and Q( 2, 2 ) are an two points on a line, then m, the slope of the line, is given b m vertical change horizontal change 2 1 when 2 2 Z 1 1 This definition provides eactl the numerical measure of steepness that we want. If a line rises as we move from left to right, the slope will be positive the steeper the line, the larger the numerical value of the slope. If the line falls from left to right, the slope will be negative. Let s proceed to some eamples. Eample 1 Finding the Slope Find the slope of the line containing points with coordinates (1, 2) and (5, 4). 519

2 520 CHAPTER 6 AN INTRODUCTION TO GRAPHING Let P( 1, 1 ) (1, 2) and Q( 2, 2 ) (5, 4). B the definition of slope, we have m (5, 4) (1, 2) (5, 2) Note: We would have found the same slope if we had reversed P and Q and subtracted in the other order. In that case, P( 1, 1 ) (5, 4) and Q( 2, 2 ) (1, 2), so m It makes no difference which point is labeled ( 1, 1 ) and which is ( 2, 2 ). The resulting slope will be the same. You must simpl sta with our choice once it is made and not reverse the order of the subtraction in our calculations. CHECK YOURSELF 1 Find the slope of the line containing points with coordinates (2, 3) and (5, 5). B now ou should be comfortable subtracting negative numbers. Let s appl that skill to finding a slope. Eample 2 Finding the Slope Find the slope of the line containing points with the coordinates ( 1, 2) and (3, 6). Again, appling the definition, we have m 6 ( 2) ( 1) ( 1, 2) (3, 6) 6 ( 2) 8 (3, 2) 3 ( 1) 4

3 THE SLOPE OF A LINE SECTION The figure below compares the slopes found in the two previous eamples. Line l 1, from 1 Eample 1, had slope. Line l 2, from Eample 2, had slope 2. Do ou see the idea of slope 2 measuring steepness? The greater the slope, the more steepl the line is inclined upward. l 2 m 2 l 1 1 m 2 CHECK YOURSELF 2 Find the slope of the line containing points with coordinates ( 1, 2) and (2, 7). Draw a sketch of this line and the line of Check Yourself 1. Compare the lines and the two slopes. Let s look at lines with a negative slope. Eample 3 Finding the Slope Find the slope of the line containing points with coordinates ( 2, 3) and (1, 3). B the definition, m ( 2) 3 2 ( 2, 3) m 2 (1, 3) This line has a negative slope. The line falls as we move from left to right. CHECK YOURSELF 3 Find the slope of the line containing points with coordinates ( 1, 3) and (1, 3).

4 522 CHAPTER 6 AN INTRODUCTION TO GRAPHING We have seen that lines with positive slope rise from left to right and lines with negative slope fall from left to right. What about lines with a slope of zero? A line with a slope of 0 is especiall important in mathematics. Eample 4 Finding the Slope Find the slope of the line containing points with coordinates ( 5, 2) and (3, 2). B the definition, m ( 5) ( 5, 2) m 0 (3, 2) The slope of the line is 0. In fact, that will be the case for an horizontal line. Because an two points on the line have the same coordinate, the vertical change 2 1 must alwas be 0, and so the resulting slope is 0. CHECK YOURSELF 4 Find the slope of the line containing points with coordinates ( 2, 4) and (3, 4). Because division b 0 is undefined, it is possible to have a line with an undefined slope. Finding the Slope Find the slope of the line containing points with coordinates (2, 5) and (2, 5). B the definition, m Eample 5 5 ( 5) Remember that division b zero is undefined. (2, 5) An undefined slope (2, 5)

5 THE SLOPE OF A LINE SECTION We sa that the vertical line has an undefined slope. On a vertical line, an two points have the same coordinate. This means that the horizontal change 2 1 must alwas be 0 and because division b 0 is undefined, the slope of a vertical line will alwas be undefined. CHECK YOURSELF 5 Find the slope of the line containing points with the coordinates ( 3, 5) and ( 3, 2). Given the graph of a line, we can find the slope of that line. Eample 6 illustrates this. Eample 6 Finding the Slope from the Graph Find the slope of the line graphed below. We can find the slope b identifing an two points. It is almost alwas easiest to use the and intercepts. In this case, those intercepts are (3, 0) and (0, 4). Using the definition of slope, we find m 0 ( 4) The slope of the line is. 3 CHECK YOURSELF 6 Find the slope of the line graphed below.

6 524 CHAPTER 6 AN INTRODUCTION TO GRAPHING In Section 6.3, we saw that a line could be drawn from two ordered pairs. Given equations of the form k, it is fairl eas to find two ordered pairs. In the net eample, we will use those ordered pairs to find the graph of the equation. Eample 7 Graphing an Equation of the Form k (a) Find the graph of the equation 2. From the table to the right, we know that the ordered pairs (0, 0) and (1, 2) are solutions to the equation The graph is displaed below Note that the slope of the line that passes through the points (0, 0) and (1, 2) is m 0 ( 2) Find the graph of the equation 1. 3 From the table to the right, we know that the ordered pairs (0, 0) and (3, 1) are solutions to the equation

7 THE SLOPE OF A LINE SECTION The graph is displaed below Note that the slope of the line that passes through the points (0, 0) and (3, 1) is m CHECK YOURSELF 7 Find the graph of the equation 1 2. In Eample 7, we noted that the slope of the line for the equation 2 is 2, and the 1 slope of the line for the equation 1 is. This leads us to the following observation. 3 3 The slope of a line for an equation of the form k will alwas be k. Because k is the slope, we generall write the form as m Note that (0, 0) will be a solution for an equation of this form. As a result, the line for an equation of the form m will alwas pass through the origin. The following sketch summarizes the results of our previous eamples. The slope is undefined. m is positive. NOTE As the slope gets closer to 0, the line gets flatter. m is 0. m is negative. Four lines are illustrated in the figure. Note that 1. The slope of a line that rises from left to right is positive. 2. The slope of a line that falls from left to right is negative. 3. The slope of a horizontal line is A vertical line has an undefined slope.

8 526 CHAPTER 6 AN INTRODUCTION TO GRAPHING CHECK YOURSELF ANSWERS 1. m m 5 3 (2, 7) (5, 5) ( 1, 2) (2, 3) 3. m 3 4. m 0 5. m is undefined 6. m (Note: Your second point could have been ( 2, 1) or (2, 1).)

9 Name 6.4 Eercises Section Date Find the slope of the line through the following pairs of points. 1. (5, 7) and (9, 11) 2. (4, 9) and (8, 17) ANSWERS ( 2, 5) and (2, 15) 4. ( 3, 2) and (0, 17) ( 2, 3) and (3, 7) 6. ( 3, 4) and (3, 2) ( 3, 2) and (2, 8) 8. ( 6, 1) and (2, 7) (3, 3) and (5, 0) 10. ( 2, 4) and (3, 1) (5, 4) and (5, 2) 12. ( 5, 4) and (2, 4) 13. ( 4, 2) and (3, 3) 14. ( 5, 3) and ( 5, 2) ( 3, 4) and (2, 4) 16. ( 5, 7) and (2, 2) 17. ( 1, 7) and (2, 3) 18. ( 4, 2) and (6, 4) In eercises 19 to 24, two points are shown. Find the slope of the line through the given points

10 ANSWERS In eercises 25 to 30, find the slope of the lines graphed

11 ANSWERS Find the graph of the following equations

12 ANSWERS 37. (a) 37. Consider the equation 2 5. (a) Complete the following table: 38. (a) (a) Use the ordered pairs found in part (a) to calculate the slope of the line. What do ou observe concerning the slope found in part and the given equation? 38. Repeat eercise 37 for 3 and (a) 39. Repeat eercise 37 for 1 and (a) (d) 42. (a) (d) 40. Repeat eercise 37 for 4 6 and 41. Consider the equation: 2 3 (a) Complete the following table of values, and plot the resulting points. Point A 5 B 6 C 7 D 8 E (a) (d) 44. (a) (d) As the coordinate changes b 1 (for eample, as ou move from point A to point B), how much do the corresponding coordinates change? Is our answer to part the same if ou move from B to C? from C to D? from D to E? (d) Describe the growth rate of the line using these observations. Complete the following statement: When the value grows b 1 unit, the value. 42. Repeat eercise 41 using: Repeat eercise 41 using: Repeat eercise 41 using:

13 ANSWERS In the following eercises, (a) plot the given point; using the given slope, move from the point plotted in (a) to plot a new point; draw the line that passes through the points plotted in (a) and. 45. (3, 1), m ( 1, 4), m a. b. c. d. e. f. 47. ( 2, 1), m ( 3, 5), m 2 Getting Read for Section 6.5 [Section 2.2] Solve each equation for. (a) (d) 14 3 (e) 24 9 (f ) Answers Undefined

14 (a) (3, 1), (4, 3); 2; slope equals coefficient of (a) (3, 1), (6, 0); 1 ; slope equals 3 coefficient of 41. (a) (5, 13), (6, 15), (7, 17), (8, 19), (9, 21); 2; Yes; (d) increases b (a) (5, 30), (6, 26), (7, 22), (8, 18), (9, 14); 4; Yes; (d) decreases b a. 5 b. 3 c. 7 d. e. 8 f