1.4
Equations Section of Lines and Modeling
Consider the three linear equations below:
a. y 4 x
3
b. y 4 x 5
3
c. y 4 x 3
3
W hat are the slopes and y-intercepts of these lines?
These equations are written in Slope-Intercept form: y mx b
where m is the slope and b is the y-intercept.
Enter these 3 equations into your calculator as Y1, Y2, and Y3
and draw the graph.
Sketch the graph
from your screen:
These graphs illustrate a very important fact:
Non-vertical lines are parallel if and only if they have the same
slope and different y-intercepts.
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Math 1513 Sec 1.4
Now enter this equation for Y4: y 3 x 2
4
This last equation has a special relationship to the other 3 lines.
The first 3 lines have a slope of 4 3 and the last equation has a
slope of 3 4 and (4 3)(3/4) 1.
This example is suppose to illustrate another fact:
Two lines with slopes m1 and m2 are perpendicular if and only
if the product of their slopes is -1. That is, m1 m2 1.
The fourth line we graphed is supposed to be perpendicular to
the other three lines. However, it doesnt look like it.
W hats wrong? What could we do on the calculator to make
the lines look perpendicular?
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Math 1513 Sec 1.4
Finding equations of lines using the Point-Slope form.
a. Find the equation of the line with a slope of 1 and passing
2
through the point (6, 5).
b. Find the equation of the line that passes through the point
(-3, 2) and is perpendicular to the line 5 y 4 x 3 .
c. Find the equation of the line that passes through the points
(-3,4) and (2,15).
y y
First, find the slope: m 2 1 15 4 11 .
x2 x1 2 (3) 5
Now we know the equation looks like y 11 x b .
5
Pick one of the points and substitute for x and y.
15 11(2) b
5
15 22 b
5
15 22 b
5
75 22 b
5
5
53 b
5
Therefore, the equation is
y 11 x 553 or 5 y 11x 53 or 5 y 11x 53 0
5
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Math 1513 Sec 1.4
There is another formula or equation that is useful in finding the
equations of lines.
Point-Slope Formula
The line with a slope of m passing through the point ( x1, y1) is
given by the equation y y1 m( x x1) .
a. Find the equation of the line passing through the points
(-3, -6) and (6, 0).
b. Find the equation of the line passing through the point (3, 9)
and parallel to the line y 5x 2 0
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Math 1513 Sec 1.4
Summary of Terminology about Lines
(p. 119)
Slope
y y
y y
m x2 x1 or m x1 x2
2
1
1
2
Slope-Intercept equation
y mx b
Point-Slope equation
y y1 m( x x1)
Horizontal line
yb
Vertical line
xa
Parallel lines
m1 m2, b1 b2
Perpendicular lines
m1 m2 1
Mathematical Models
W hen a real-world problem can be described in mathematical
language, we have a mathematical model.
Example:
Suppose the cost in dollars, to C, make q items of a product is
given by the function C (q) 1200 3q .
a. What does C (0) represent and what is its value?
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Math 1513 Sec 1.4
b. What symbol would be used to represent the cost to make
100 items? What is the cost?
c. This equation is linear. What is the slope and what meaning
does the slope have?
Linear Regression
Many times in real-world applications, we do not have a nice,
neat formula to express the relationship between 2 variables.
Instead, we have a table of values of the variables.
Example:
Suppose we know the height (in inches) of a boy at various
ages of his life. This is shown in the table:
Age (years) x
Height (in.) y H ( x)
2 4 6 8 10 12
35 43 45 51 58 60
If we plotted this data, the plot would look something like this.
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Math 1513 Sec 1.4
The data points are not exactly in a straight line, but they are
close. If we took a ruler and tried to draw a line through the
points, it might look like this.
I
If each of you in this class drew the line, all of your lines would
be slightly different depending on what you thought line through
the points should be. Mathematically, there is a best line that
passes through the points.
The mathematical method used to find the equation of this line
is beyond the scope of this course. It suffices to say that this
best line minimizes the distance between the points and the
line. Fortunately, your calculator can find the equation of this
line very easily. This process is called Linear Regression.
There is a document (1.4 Lin Reg ) in the website and the title
is Regression. It contains a sample problem and step-by-step
directions for doing regression.
You should print this out and bring it to class.
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Math 1513 Sec 1.4
To find the equation of the regression line for this data,
Age (years) x
Height (in.) y H ( x)
2 4 6 8 10 12
35 43 45 51 58 60
we enter the data in a STAT table:
then we ask the calculator to find the equation:
and finally, we can graph the line with the data points:
The directions for all these steps are given in the regression
document.
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Math 1513 Sec 1.4
Suppose in the original problem, we were asked to predict how
tall the boy will be when he is 16 years old.
This screen tell us that the formula for the regression line
through the data is: y 2.514x 31.067 (rounded to 3
decimals).
Another way to think of this is that his height H at any age x is
given by the formula: H ( x) 2.514 x 31.067 .
In function notation, his height at 16 years would be H (16) so
H ( x) 2.514 x 31.067
H (16) 2.514(16) 31.067
H (16) 79.291
Therefore, if he continues to grow linearly and if our model is
still valid, when he is 16 he would be slightly taller than 79
inches.
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Math 1513 Sec 1.4