formula
me
November 2024
1
1.Secant line formula.
A secant line is a line on any graph that touches the graph at two coordinates.
First, we have the equation for the slope of a secant line.
m=
f (x2 ) − f (x1 )
x2 − x1
(1)
This is based on the formula for slope.
y2 − y1
x2 − x1
The secant version shows the numerator part as:
m=
(2)
f (x2 ) − f (x1 )
(3)
y2 − y1
(4)
To be as
This is because a function itself is equal to the y coordinate.
Now, the formula in (1) is for the slope of the secant line; the equation
would be of the form.
y = mx + c
(5)
So, the slope has already been defined in (1), now what about the x. The x
is defined as a point desired for a specific y coordinate, so, this would be written
as:
y = m(x − x1 ) + c
(6)
Here, you subtract to find the distance between the desired point and the ’fixed’
left most coordinate.
Now for the y intercept.
f (x1 ) = m(x1 − x1 ) + c
1
(7)
We use that specific coordinate to find c, because we know when x = 0, c is
found, and to make x = 0, we subtract it by itself as shown above, by inputting
a value of x equal to it, which in this case is always .
f (x1 )
(8)
f (x) = m(x1 − x) + f (x1 )
(9)
Now, the slope has been defined as:
And we earlier defined the slope, see fig (1)
So.
f (x) =
f (x2 ) − f (x1 )
(x − x1 ) + f (x1 )
x1 − x2
(10)
Remember this entire formula consists of a secant line, which is a line that
touches two points on any graph.
1.1
1.1 Secant line limit version.
The secant line slope formula described in Section 1 is the non-limit version,
the limit version is.
f (x + h − f (x))
(11)
h
Here, h is the horizontal distance between the two points on the secant line.
You add h to the input of the right most coordinate logically because if h is
the distance between the two, it makes sense to add.
Also, we divide by h since h is the distance between the two, which is basically
:
m=
x1 − x2
a
2
(12)