The linear function is popular in economics. It is attractive because it is simple and easy to handle mathematically. It has many important applications.
Linear functions
are those whose graph is a straight line.
A linear
function has the following from
y = f(x)
= a + bx
A linear
function has one independent variable and one dependent variable. The
independent variable is x and the dependent variable is y.
A is the
constant term or the y intercept. It is the value of the dependent variable
when x = 0.
b is the coefficient of the independent variable. It is
also known as the slope and gives the rate of change of the dependent variable.
Point-Slope Formula.
We have seen
that we can define the slope of a line given two points on the line, and use
than information along with the y-intercept to graph the line.
If you don’t
know the y-intercept, or the equation for the line you can use
two points to define the equation of the line using the point-slope formula.
This is an
important formula, as it will be used in other areas of college algebra and
often in calculus to find the equation of a tangent line. We need only one
point and the slope of the line to use the formula. After substituting the
slope and the coordinates of one paint into the formula, we simplify it and
write it in slope-intercept form.
Given one point
and the slope, the point-slope formula will lead to the equation of a line:
y = mx.
EXAMPLE:
Write the equation of the line with slope m = –3 and passing
through the point (4, 8). Write the
final equation in slope-intercept form.
SOLUTION:
Using the point-slope formula, substitute –3
for m
and the point (4, 8) for
(x1, y1).
y
– y1 = m(x
– x1)
y
– 8 = –3(x – 4)
y
– 8 = –3x + 12
y
= –3x + 20
Note that any point on the line can be used to find
the equation. If done correctly, the same final equation will be obtained.
EXAMPLE:
Find the equation of the line passing through the
points (3, 4) and (0, –3). Write the final equation in slope-intercept form.
SOLUTION:
First, we calculate the slope using the slope formula
and two points.
Next, we use the point-slope formula with the slope of 7/3, and either
point. Let’s pick the point (3, 4) for (x1, y1).
y – 4 = 7/3(x – 3)
y – 4 = 7/3
x – 7
y = 7/3
x – 3
Distribute the 7/3.
In slope-intercept form, the equation is written as
y = 7/3
x – 3.
To prove that either point can be used, let us use the
second point (0, –3)
and see if we get the same equation.
y – (–3) = 7/3(x – 0)
y + 3 = 7/3 x
y = 7/3 x – 3
We see that the
same line will be obtained using either point. This makes
sense because we used
both points to calculate the slope.
Lesson 17
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