MA 0090 Section 21 - Slope-Intercept Wednesday, October 31, 2018 Objectives: Review the slope of the graph of an equation in slope-intercept form. Last time, we looked at the equation Slope (1) y = 2x 4, as an example. To graph it, we found the points ( 1, 6), (0, 4), (1, 2), (2, 0), and (3, 2). These points and the line are plotted below. Note that the y-coordinates increase by 2 from one point to the next. This goes with the fact that I chose x-coordinates that increase by 1. In other words, as x increases by 1, y increases by 2. This should make sense, because the coefficient on the x-term is multiplied by 2. As the x gets 1 bigger, the 2x becomes 2 bigger, and the 4 stays the same. As a result, the y should get 2 bigger. In general, if we have a linear equation in the form (2) y = mx + b, the y will increase by m as the x increases by 1. The number m is called the slope, since that s how fast y increases. Consider the equation (3) y = 2 3 x 4. If we start looking at points starting at x = 0, we get (0, 4), (1, 10 3 ), (2, 8 3 ), (3, 6 3 ), and (4, 4 ). These 3 points are plotted below. 1
MA 0090 Section 21 - Slope-Intercept 2 Here, as the x increases in increments of 1, the y increases by 2 3. These are fractions, however, and that can often be inconvenient. Note that we do hit whole number coordinates at (3, 2). That means that as x increases by 3, y increases by 2. The fraction m = 2 3 still makes sense (4) m = 2 3 = up 2 right 3. right 3 up 2 We could just as easily interpreted the slope for this line as (5) m = 2 3 = 2 3 = down 2 left 3. This is how you would go from (3, 2) to (0, 4). If you interpret negative y s as going down, and negative x s as going to the left, any way you write the slope will tell you how to go from any point on the line to another point on the line. Quiz 21, Part I Interpret each slope like 2 3 = down left 3 2. In D2L, you ll enter this as down2/left3 with no spaces. 1. m = 1 3. 2. m = 1 3 = 1 3. 3. m = 2 3 = 2 3. 4. m = 2 3 = 2 3. 5. m = 2 3 = 4 6.
MA 0090 Section 21 - Slope-Intercept 3 y-intercepts and graphing When we re graphing a linear equation, we really only need to know two points, since we already know that the graph is going to be a straight line. For the equation (6) y = 2 3 x 4 that we graphed before, we can find two points really easily. One point is the y-intercept. Putting x = 0 into the equation gives us y = 4 (i.e., the point (0, 4)). The y-intercept is the same as the constant term, when the equation is written in this form. We now need a second point. We can get this other point using the slope (7) m = 2 3 = up 2 right 3. right 3 up 2 For an equation in this form, therefore, we can graph the line very quickly using the y-intercept and the slope. This form (8) y = mx + b is called the slope-intercept form of a line. The coefficient of the x-term is the slope, m, and the constant term is the y-intercept, b. Let s do a fresh example. Graph y = 3 4x + 2. The y-intercept is b = 2, and the slope is (9) m = 3 4 = 3 4 = down 3 right 4. One point that we can use, therefore, is (0, 2), and then we can count over right 4 and down 3 (or equivalently, down 3 and right 4) to get the second point. right 4 down 3
MA 0090 Section 21 - Slope-Intercept 4 Quiz 21, Part II To graph the following linear equations, you ll need to plot two points. Find the coordinates of the y-intercept (like(0,2)), and the second point you got from using the given interpretation of the slope (like(3,2)). D2L will ask for these two points separately. 6. y = 4 3 x 2 (use m = 4 3 7. y = 2x 3 (use m = 2 1 to find the second point.) to find the second point) Homework starts on next page
MA 0090 Section 21 - Slope-Intercept 5 Homework 21 Interpret each slope like 2 3 = down left 3 2. Write your answer in D2L in the form down2/left3 with no spaces. 1. m = 4 3. 2. m = 4 3 = 8 6. 3. m = 2 5 = 2 5. 4. m = 2 5 = 2 5. 5. m = 2 5 = 4 10. For problems 6-9, consider the equation y = 5 2 x + 3. 6. What are the coordinates of the y-intercept? Write your answer like (0,-5) with no spaces. 7. What are the coordinates of your second point using the slope m = 5 2? 8. What would the coordinates of your second point be, if you used the slope m = 5 9. Would your answers from problems 7 and 8 give you the same line, if you used them to graph the equation? For problems 10-13, consider the equation y = 3x 2. 10. What are the coordinates of the y-intercept? 11. What are the coordinates of your second point using the slope m = 3 1? 12. What would the coordinates of your second point be, if you used the slope m = 6 13. Are the slopes from problems 11 and 12 both OK? For problems 14-17, consider the equation y = 1 4 x + 1. 14. What are the coordinates of the y-intercept? 15. What are the coordinates of your second point using the slope m = 1 4? 16. What are the coordinates of your second point using the slope m = 1 4? 17. Are the slopes from problems 15 and 16 both OK? For problems 18-20, consider the equation y = 2x + 5. 18. What are the coordinates of the y-intercept? 19. What are the coordinates of the second point using the slope m = 4 20. Is the slope in problem 19 OK to use? (even if it s not the simplest choice)