The table shows the closing value of a stock index for one week in March, 2004.
a. Using the day as the x-value and the closing value as the y-value, write equations in slope-intercept form for the lines that represent each value change.
b. What would indicate that the rate of change for two pair of days was the same? Was the rate of change the same for any of the days shown?
c. Use each equation to predict the closing value for the next business day (Day 22). The actual closing value was 1909.90. Did any equation correctly predict this value? Explain.
Stock Index
March, 2004
______________________
Day - Closing Value
15 - 1939.20
16 - 1943.09
17 - 1976.76
18 - 1962.44
19 - 1940.47
for each pair do
y = m x + b
for example for day 15 and 16
1939.2 = m (15) + b
1943.9 = m (16) + b
------------------------ subtract
- 4.7 = -1 m
so
m = 4.7 - slope That is the "rate of change" for part b
back for b value
1939.2 = (4.7)(15) + b
b = 1868.7
so for the first two points the line is
price = 4.7*day + 1868.7
for the day 22
price = 4.7*22 + 1869.7
= 1973.1
The actual price on day 22 was 1909.9
There is no reason any combination of the prices of the previous week should predict the price on Monday. It happened that the price was going up fast between Monday and Tuesday. All we just did was assume that it continued to go up like that for a week, which it did not do.
thanks
a. To write equations in slope-intercept form for the lines that represent each value change, we need to first find the slope (m) and the y-intercept (b) for each line.
For the first pair of days (Day 15 and Day 16):
Slope (m) = (change in y)/(change in x) = (1943.09 - 1939.20)/(16 - 15) = 3.89
Using the slope-intercept form y = mx + b, we can substitute the values of one of the points (e.g., Day 15) to find b:
1939.20 = 3.89 * 15 + b
b = 1939.20 - 3.89 * 15
b ≈ 1933.75
So, the equation for this line is y = 3.89x + 1933.75.
For the second pair of days (Day 16 and Day 17):
Slope (m) = (change in y)/(change in x) = (1976.76 - 1943.09)/(17 - 16) = 33.67
Using the slope-intercept form y = mx + b, we can substitute the values of one of the points (e.g., Day 16) to find b:
1943.09 = 33.67 * 16 + b
b = 1943.09 - 33.67 * 16
b ≈ 1386.57
So, the equation for this line is y = 33.67x + 1386.57.
For the third pair of days (Day 17 and Day 18):
Slope (m) = (change in y)/(change in x) = (1962.44 - 1976.76)/(18 - 17) ≈ -14.32
Using the slope-intercept form y = mx + b, we can substitute the values of one of the points (e.g., Day 17) to find b:
1976.76 = -14.32 * 17 + b
b = 1976.76 + 14.32 * 17
b ≈ 2202.08
So, the equation for this line is y = -14.32x + 2202.08.
For the fourth pair of days (Day 18 and Day 19):
Slope (m) = (change in y)/(change in x) = (1940.47 - 1962.44)/(19 - 18) ≈ -21.97
Using the slope-intercept form y = mx + b, we can substitute the values of one of the points (e.g., Day 18) to find b:
1962.44 = -21.97 * 18 + b
b = 1962.44 + 21.97 * 18
b ≈ 2067.98
So, the equation for this line is y = -21.97x + 2067.98.
b. The rate of change for two pairs of days is the same if the slopes (m) of their equations are equal. Let's compare the slopes:
Slope between Day 15 and Day 16: 3.89
Slope between Day 16 and Day 17: 33.67
Slope between Day 17 and Day 18: -14.32
Slope between Day 18 and Day 19: -21.97
We can see that the rate of change is not the same for any of the days shown because the slopes are all different.
c. We can use each equation to predict the closing value for the next business day (Day 22) by substituting x = 22 into the equations.
Using the equation for the first pair of days (y = 3.89x + 1933.75):
y = 3.89 * 22 + 1933.75
y ≈ 2023.33
Using the equation for the second pair of days (y = 33.67x + 1386.57):
y = 33.67 * 22 + 1386.57
y ≈ 2123.71
Using the equation for the third pair of days (y = -14.32x + 2202.08):
y = -14.32 * 22 + 2202.08
y ≈ 1949.64
Using the equation for the fourth pair of days (y = -21.97x + 2067.98):
y = -21.97 * 22 + 2067.98
y ≈ 1597.44
The actual closing value for Day 22 was 1909.90. None of the equations correctly predicted this value. The predicted values using the equations were: approximately 2023.33, 2123.71, 1949.64, and 1597.44.