Slope of a Line



Slope of a Line (Solutions)

Goal: Students should be able to complete a table, graph a line, write an equation, analyze a graph, and recognize vital parts of the equation

y = mx + b.

Students should recall that slope is calculated by picking any two points on the line (x1 , y1) and (x2, y2). Divide the rise (difference between the y-coordinates) by the run (difference between the x-coordinates).

Slope = y2 - y1 = rise

x2 – x1 run

Students should also recall that a positive slope rises from left to right, a negative slope falls from left to right, a slope of zero is a horizontal line, and a vertical line has an undefined slope.

1. Complete the table and then graph the line for the equation y = x – 4.

|x |x - 4 |y f(x) | (x,y) |

|Choose a value for x |Substitute the value of x |Find the value of y |Create an ordered pair |

|5 |1 |1 |(5,1) |

|2 |-2 |-2 |(2,-2) |

|0 |-4 |-4 |(0,-4) |

|8 |4 |4 |(8,4) |

|[pic] |

|Figure 1 |

2. Complete the table and draw the graph of the equation y = -x – 3..

|x | - x – 3 |y f(x) | (x,y) |

|1 |-4 |4 |(1,4) |

|2 |-5 |5 |(2,5) |

|3 |-6 |6 |(3,6) |

|4 |-7 |7 |(4,7) |

|[pic] |

| |

3. Construct and complete a table, then graph the equation of y = 7 – 2x.

|[pic] |

4. Using a graphing calculator, graph the equation y = 1/2x + 1. Use the window (xmin = -10, xmax = 10, xscl = 1, ymin = -10, ymax = 10, yscl -= 1, xres = 1). Enter the equation into y= (upper left corner) and then graph (upper right corner). Sketch your findings below.

5. What are the slopes for problems 1, 2, 3, 4? What correlation do you see between these slopes and their respective equations?

1,-1,-2,1/2

6. Are the slopes for problems 1, 2, 3, 4 positive or negative? What correlation do you see between the signs of the slopes and their respective equations?

Positive,negative,negative,positive

7. What are the y-intercepts for problems 1, 2, 3, 4? What correlation do you see between their y-intercepts and their respective equations?

-4,-3,7,1

8. Graph the equations y = 2x, y = 2x + 1, y = 2x + 2, y = 2x + 10, and y = 2x – 3.

9. What do you notice about the change in the graphs in relation to their respective equations?

Steeper with bigger slopes, positive with pos slopes, negative with negative slopes

10. Graph the equations y = x + 3, y = 2x + 3, y = 3x + 3, y = 10x + 3, and

Y = -2x + 3.

11. What can you conclude about the relationship between the graphs and their respective equations?

These two things hold a direct relationship they are the same thing only in different physical representations

12. Given this graph, what is its slope? What is its y-intercept?

|[pic] |

½=m y=2

13. Given this graph, what is its slope? What is its y-intercept?

|[pic] |

y=0 m=-1/3

14. Given this graph, what is its slope? What is its y-intercept?

|[pic] |

m=-1 y=-2

15. Given this graph, what is its slope? What is its y-intercept?

|[pic] |

m=0 y=2

16. What are the equations for problems 11, 12, 13, 14? How do you know these equations correctly match their graphs?

12. y=x/2+2

13. y=-x/3

14. y=-x-2

15. y=2

Scatter plots show the relationship between two sets of data. The two sets of data are graphed as ordered pairs on a coordinate system. When real-life data are collected, the points graphed usually do not form a straight line, but may approximate a linear relationship. Scatter plots can be used to spot trends, draw conclusions, and make predictions about the data.

The relationships between the two sets of data can show a positive relationship where y increases as x increases. The plots can show a negative relationship where y decreases as x increases. There can also be no relationship if that data reveals no obvious relationship.

A best-fit line is used with a scatter plot to approximate a linear relationship. A best-fit line is a line that is very closed to most of the data points.

17. Use the table to make a scatter plot and draw a best-fit line. Use the best-fit line to predict the average hourly wage of construction workers in 2010.

|Year |Average Hourly Earnings ($) |

|1980 |9.94 |

|1985 |12.32 |

|1990 |13.77 |

|1995 |15.09 |

|2000 |17.13 |

|[pic] |

18. Use the table to make a scatter plot and draw a best-fit line. Then write an equation for the best-fit line and use it to predict the number of persons employed in mining in 2010.

|Year |Employees |

| |(thousands) |

|1980 |1027 |

|1985 |927 |

|1990 |709 |

|1995 |581 |

|2000 |535 |

|[pic] |

19. What concepts have you learned?

How to go from working graphically to working numerically.

How to work numerically to reversing to graphically

How to find slope

How to transfer the table values to the graph coordinates

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