Math 25 Activity 8: Plotting Points, Lines, Slope, and Y-intercept This week we are have two parts for the activity. PART ONE: In this part of the activity, we will review the Cartesian coordinate system and review plotting points. Imagine you live in the North End at the corner of 15 th and Eastman: 1. How many different ways could you walk to the corner of Ridenbaugh and 12 th? 2. Is there an efficient way to describe all the routes you found so that everyone could follow what you mean? 3. If there are infinitely many routes from one corner to another, then can you think of a circumstance when you would only want one route? Let s look at a short reminder for plotting points on a Cartesian coordinate system. The coordinate system is based on taking ordered pairs (which we call points when they are plotted) and interpreting them as directions from the origin, where the x-axis crosses the y-axis. Directions from the ordered pair, (x,y), mean we move the point x units right from the origin when the number in Page 1 of 7
the x-coordinate is positive and x units left from the origin when the number is negative. Then from there, we move y units up from that point if the y-coordinate number is positive or y units down if the number is negative. The axes should always be labeled with a scale or the plotted point has no significance. The scale should also be a consistent distance so that a unit remains the same amount for each mark. When the points lie on an axis we call them intercepts. When they are not located on an axis, we say that they lie in a quadrant. The quadrants are labeled counter-clockwise starting with the upper-right quadrant, which is where both axes are labeled with positive numbers. Now look at the map from the first page. Make your home the origin by darkening the roads 15 th and Eastman. Eastman becomes your x-axis and 15 th street becomes your y-axis. Using the map and your home as the origin: 4. What ordered pair represents the corner of Ridenbaugh and 12 th? 5. What ordered pair represents the corner of 18 th and Ada? 6. Did you label the scale of each street before recording the ordered pairs? Why does this matter? What does it mean in terms of maps and distance? 7. Does this map meet the requirements for a Cartesian coordinate system? Why or why not? 8. What cross streets are associated with the ordered pair (-3,3)? 9. How would the activity change if you lived at the corner of 11 th and Lemp? Page 2 of 7
Instructor! Your instructor will pause to discuss some of the main points from Part One. PART TWO: In this part of the activity we will explore graphing lines, slope, and y-intercept. Imagine you have a marble on a track. For our purposes, the track is 2-dimensional (just upward track or downward track) and made of mostly lines except for small curves when changing direction (no loops). Draw a track that has parts where the marble is travelling up and parts where it is travelling down. Also, make some of the inclines more steep than other parts, similarly draw the declines (see the next page for an example but try to draw your own example here). 10. Which are the steepest inclines and declines on the track you drew? 11. Which are the least steep inclines and declines on the track you drew? Page 3 of 7
Answer the same questions for this hand-drawn example of a track. 12. Which are the steepest inclines and declines on the above track? 13. Which are the least steep inclines and declines on the above track? The slope of the line is a numerical value, usually indicated by the letter m, that indicates the steepness of a line and whether the line looks like it is declining or inclining. Visually you determined that there is a difference in steepness and a difference between incline and decline, but let s explore how to tell the difference based on the numerical value that is the slope. Consider the following graphed line. We can make a right triangle by using the points indicated on the graph (-3,7) and (6,4). 14. How many units is the height of the triangle (we call this the rise because it is vertical)? 15. How many units is the length of the triangle (we call this the run because it is horizontal)? 16. What numerical value do you get when you take the rise and divide it by the run? Page 4 of 7
Now, consider (0,6) which is a third point on the same line as seen on the previous page. We repeat the process making a new right triangle. 17. How many units is the height of the new triangle (rise)? 18. How many units is the length of the new triangle (run)? 19. What numerical value do you get when you take the rise and divide it by the run? 20. How does this number compare with your answer to question 16? The slope of the line is defined to be the constant number calculated by taking any two unique points on a line and dividing the rise by the run. We also want to note here that because when we look at the graph, if this line were a track, then the marble would roll downhill. Consider moving your finger along the legs of the triangle in the first example. You could have started at the point (-3,7) and moved down 3 units, then right 9 units. The word down really relates to a negative value for the rise while the word up relates to a positive rise. The word right relates to a positive value for the run while the word left relates to a negative run. The slope you calculated in question 5, which is rise divided by run, should be negative because you would divide -3 by 9 which reduces to the fraction -1/3. 21. Describe the directions if you traced the triangle starting at the point (6,4) and traveled to (0,6) using the words up/down and left/right. 22. Do you still come up with a negative slope starting at a different point? Page 5 of 7
23. Use the definition of slope to find the slope of the following line. Draw the right triangle you used to help you. The formula for slope is m y x 2 2 y1 x 1 where you label your points how you want, your first point would be x, ) and your second point would be x, ). The subtraction is what calculates the ( 1 y1 ( 2 y2 difference between the points for both the rise and the run. Notice how the change in y values is in the numerator of the fraction and the change in x values is in the denominator. Mathematically, this symbolizes rise divided by run. The formula also accounts for the slope being negative or positive. If the number comes out negative, then you should see a decline. If the number comes out positive, then you should see an incline. 24. Draw a line on the graph below that is more steep than the given line that has a positive slope and draw a second line that is less steep than the given line but still has a positive slope. Calculate all three slopes. Page 6 of 7
25. What pattern do you notice when you compare the steepness of the line and the slope? When you have positive slopes, the closer the slope is to zero, the less steep the line. The larger the slope the more steep the line appears. When you have negative slopes, the closer the slope is to zero, the less steep the line. The more negative a number, the steeper the slope of the declining line. The last thing we want to consider about graphing lines in this project is the difference between lines with the same slope. Slope helps us imagine what the line might look like on a blank piece of paper, but when graphing lines there is one more important factor that we need to consider. 26. The following lines have the same slope. What is unique about each line that can distinguish it from the other two lines? There are multiple correct answers to question 26, but mathematicians have decided to use the point where a line crosses the y-axis to be the second piece of pertinent information to graph a line. We call it the y- intercept and it is represented using the letter b. Circle the three y-intercepts and determine their values. Therefore, after exploring the slope and the y-intercept, you have the tools to write the equation of a line using the slope-intercept formula y=mx+b. You should also be able to take an equation in that form and graph it. Page 7 of 7