DAY 4 SLOP-INTERCEPT FORM EXAMPLE 1 The equation of the line that represents Kims skating costs is . how can you use the slope-intercept formula to construct the graph?
ANSWER The formula is Since b is 50, measure 50 units up from the origin on the , 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 , such as or . Move right
10, then up 30. Graph the second point. Draw a line through the points. 30 10 Why do you think the line in Kims 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. 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. SLOPE FORMULA Given two points with coordinates and the formula for the slope is . EXAMPLE 2
Kims 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? 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 . 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 . The equation for the line is . 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). 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 74.59 miles per hour to 185.987 miles per hour in 1990. 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 1915 to 1975 in 5-year 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, 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, yintercept, and the equation, based on the line of best fit. L1 15 20 25 L2
89.84 88.62 101.1 3 30 100.4 5 35 106.2 4 40 114.2 Place the8 information
from table into a 50 the124.0 2-variable 0 (x, y) data table LinReg 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. The calculator will graph the scatter plot and draw the line of best fit when you enter the equation for the regression line after . The rate of change in average is the slope,
about 1.16 miles per hour per year. The line of best fit (regression line) has the approximate equation . 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. Hours on the job Pay for the
job 3 5 7 10 15 $28.8
0 $48 $67.2 0 $96 $1.44 Write an equation that will calculate the hourly wage that Jill earns.
ANSWER In looking at Jills records, as the number of hours she works increases, the pay she earns also increases at the same rate. This is an example of The pay that Jill earns varies directly as the number of hours she works. The hourly wage that Jill earns is determined by Hourly wage, so $9.60. The hourly wage, $9.60, is a constant.
ANSWER Let represent the pay for the job. Let represent the hours on the job. If you solve for , you find that The equation models direct variation. The expression is a ratio, and 9.6 is the constant of variation.
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