4.3
Point-Slope Form and Linear Regression
Homework: p. 234; Odds 1-7, Odds
15-29, 31, 35, 41, 45
| Nancy J. McCormick's Home Page | DSPM 0800 Class Notes |
Point-Slope Form:
In this form of the equation, we can see the slope and a point on the line,
where
This form of the equation is useful when we are asked to find the equation of
the line.
Example: Find
the equation of the line passing through the point (-5, 6) with slope 3.
Write Point-Slope form.
Plug in the slope and the point coordinates.
Simplify the equation.
This is the point-slope equation.
Remove the parenthesis.
Add 6 to both sides to get y by itself.
This is the slope-intercept equation.
or
This is the linear function.
Example: Find the equation of the line passing through the point (4, -2) with slope 2/3.
Write Point-Slope form.
Plug in the slope and the point coordinates.
Simplify the equation.
This is the point-slope equation.
Remove the parenthesis.
Subtract 2 to get y by itself.
This is the slope-intercept equation.
or
This is the linear function.
We
have seen that to find the equation of a line, we need a point on the line and
the slope of the line. Suppose we are only given two points on the line, can
we still find the equation of the line? Yes, first we can use the two points
and the slope ratio formula to find the slope. Then we use the slope that we
found plus the coordinates from one of the points (either point) and plug in
to the point-slope equation as before.
Example: Find
the equation of the line passing through the points (2, 6) and (4, 1).
Write the slope ratio.
Plug in the coordinates of the points appropriately.
Simplify the ratio.
Write the Point-Slope form.
Plug in the slope and the point coordinates.
Remove the parenthesis.
Add 6 to get y by itself.
#32, page 235:
In 1930, the record for the 1500-meter run was 3.85 min. In 1950, it was 3.70
min. Let R = the record in the 1500-meter run and t the number of years since
1930.
a) Find a linear function R(t) that fits the data.
We are given the name of the function. It is called R(t). The independent
variable is t and t is the number of years since 1930.
We need to read the problem and get two points in the form (t, R(t)).
Lets go about it by making a table of the data.
Year t (yrs since 1930) Record
1930 0 3.85
1950
20
3.70
From the table, two points of data
are: (0, 3.85) and
(20, 3.70)
Use the two points to plug in the slope ratio and find the slope of the linear
function.
Notice
that one of the points (0, 3.85) is the vertical-intercept. We know this
because the first coordinate in the point is 0.
We can use the slope and the vertical-intercept, plug into the Slope-Intercept
form and write linear function for the data, using the function notation we
were given.
R(t) = -0.0075 t + 3.85
b) Use the function of part (a) tor predict the record in 1998.
In 1998, t = 1998-1930 = 68
Substitute 68 for t in the
function.
R(68) = -0.0075 (68) + 3.85
R(68) = -0.51 + 3.85
R(68) = 3.34 minutes
You predict the record in 2002.
c) When will the record be 3.3 minutes?
This question is giving us the Record and we have to substitute 3.3 in place
of R(t) and find t.
3.3 = -0.0075 t + 3.85
3.3 3.85 = -0.0075 t + 3.85 3.85
-0.55 = -0.0075 t
74.666666.... = t
Rounding,
t is about 75. t is the years since 1930, so 1930 + 75 = 2005.
We can predict that the record will be 3.3 minutes in 2005.
#46,
p. 237:
The given table (p. 237) shows the amounts of nonfood product waste, in pounds
per person per day, for various years.
a)
Use linear regression to find a linear function that can be used to predict
the amount of nonfood product waste as a function of the year, (Let x = the
number of years since 1960).
Year x (number of years since 1960) Waste (y)
1960 0 1.65
1970 10 2.26
1980 20 2.57
1988
28
2.94
We
will use the graphing calculator to make a scatterplot of given data and use
linear regression to find a linear function.
Clear Lists. Enter the data by editing the lists.
Make Scatterplot. Turn on Plot1.
Enter ZoomStat in Zoom window.
Find the linear regression. Go to STAT, move right arrow to CALC, scroll down to #8.
Enter twice to get the equation as shown.
So the linear function used to predict the amount of nonfood product waste is:
y = .0445 x + 1.7092 (Rounding to 4 decimal places).
f(x) = .0445 x + 1.7092
b) Predict the amount of nonfood product waste per person per day in the U.S.
in 2005.
x = years since 1960 = 2005 1960 = 45
f(45) = .0445 (45) + 1.7092
f(45) = 2.0025 + 1.7092
f(45) = 3.7117 pounds per person per day
