1. Day 4 – Slop-Intercept Form

2. Example 1
The equation of the line that represents Kim’s skating costs is 𝑦=3𝑥+50. how can you use the slope- intercept formula to construct the graph?

3. Answer
The formula is 𝑦=𝑚𝑥+𝑏. Since b is 50, measure 50 units up from the origin (0, 0) on the 𝑦−𝑎𝑥𝑖𝑠, and graph the point. From that point, measure the run and the rise of the slope to locate a second point. The slope, m, is 3. you can use any equivalent of 3 1 , such as 6 2 or Move right 10, then up 30. Graph the second point. Draw a line through the points.

4. 30 10

5. Why do you think the line in Kim’s graph does not extend to the left of the y-axis? How far do you think the line would realistically extend in the positive direction? What are the reasonable domain and range? The x-values to the left of the y-axis represent negative numbers of practice hours. The graph may extend as far as (624, 1922). 624 is the number of hours Kim would skate in one year if she skated 12 hours per week for 52 weeks.

6. Points to an Equation
When you know two points on a line, you can write the equation for that line. First, calculate the slope, m, using the slope formula. Then calculate b from the slope- intercept formula and one of the points.

7. Slope Formula
Given two points with coordinates 𝑥 1 , 𝑦 1 and 𝑥 2 , 𝑦 2 the formula for the slope is
𝑚= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑥 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 2 .

8. Example 2
Kim’s notes show that after the first 7 hours of practice, the total cost for her skating that year is $71. Earlier in her notes it shows that after 5 hours of practice the total cost for the year was $65. How can you write an equation for a line knowing only this information?

9. Answer
You can represent the data as points by writing the hours and cost as ordered pairs, (5, 65) and (7, 71). To write the equation of a line from these two points, substitute the values into the slope formula. Substitute 3 for m in 𝑦=𝑚𝑥+𝑏.
𝑚= 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑥
𝑚= 71−65 7−5 − 6 2 =3
𝑦=3𝑥+𝑏

10. Answer
Next, choose either point, and substitute the coordinates for x and y into the equation. If you use the point (5, 65), substitute 5 for x and 65 for y. Then solve for b. Now substitute 3 for m and 50 b in 𝑦=𝑚𝑥+𝑏. The equation for the line is 𝑦=3𝑥+50.
𝑦=3𝑥+𝑏
60=3 5 +𝑏
65=15+𝑏
50=𝑏

11. Answer
Why is the equation in Example 2 the same as the equation in Example 1? The points (7, 71) and (5, 65) are on the same line as the line in Example 1. Write an equation for a line passing through points (3, 3) and (5, 7). 𝒚=𝟐𝒙−𝟑

12. Example 3
For decades the Indianapolis 500 automobile race has attracted the attention of millions of enthusiasts. Since 1911 the average speed for the race has increase from miles per hour to miles per hour in How steadily has this average increased? Hoe can you find an equation for the line of best fit that shows the trend in average speed? Use the data given for a sample of speed averages from to 1975 in 5-year intervals. Consider as year 0.

13. Answer
One way to find the equation for the line of best fit is to plot the points on a graph and estimate the location of the line using a clear ruler. Find the slope and y-intercept. Then substitute the numbers into the formula for a line, 𝑦=𝑚𝑥+𝑏.

14. Answer
The development in calculator technology have made graphing a scatter plot much easier. On a graphics calculator line of best fit is called regression line. The calculator will automatically calculate the slope, y- intercept, and the equation, based on the line of best fit.

15. L1 L2 15
89.84 20
88.62 25
101.13 30
100.45 35
106.24 40
114.28 50
124.00
LinReg
𝑦=𝑎𝑥+𝑏
𝑎=1.16
𝑏=68.24
𝑟=0.98
Use the linear regression feature to find the line of best fit. Here, the slope for the line of best fit is a, and y-intercept is b. The r is the correlation coefficient.
Place the information from the table into a 2-variable (x, y) data table

16. The calculator will graph the scatter plot and draw the line of best fit when you enter the equation for the regression line after 𝒚= .

17. The rate of change in average is the slope, about 1
The rate of change in average is the slope, about miles per hour per year. The line of best fit (regression line) has the approximate equation 𝑦=1.16𝑥

18. Example 4
Jill works as a cable technician and charges by the hour. Her records show the hours she works and pay she receives for various jobs. Write an equation that will calculate the hourly wage that Jill earns.
Hours on the job 3 5 7 10 15
Pay for the job
$28.80
$48
$67.20
$96
$1.44

19. Answer
In looking at Jill’s records, as the number of hours she works increases, the pay she earns also increases at the same rate. This is an example of 𝑑𝑖𝑟𝑒𝑐𝑡 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛. The pay that Jill earns varies directly as the number of hours she works. The hourly wage that Jill earns is determined by 𝑃𝑎𝑦 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑗𝑜𝑏 𝐻𝑜𝑢𝑟𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑗𝑜𝑏 = Hourly wage, so 48 5 = $9.60. The hourly wage, $9.60, is a constant.

20. Answer
Let 𝑝 represent the pay for the job. Let ℎ represent the hours on the job. 𝑝 ℎ =9.6 If you solve for 𝑝, you find that 𝑝=9.6ℎ The equation 𝑝 ℎ =9.6 models direct variation. The expression 𝑝 ℎ is a ratio, and 9.6 is the constant of variation.