What is the function f(x) which has the graph of the segment AB A(-1,-3), B(3,7) ?

We'll write the form of the linear
function:

f(x) = ax +
b

or

y = mx + n, where m
represents the slope of the line and n represents the y
intercept.

In this case, the function f(x) has as graph the
line AB.

According to the rule, a point belongs to a line
if the coordinates of the point verify the equation of the
line.

A is on the line y = ax+b if and only if yA = a*xA +
b

We'll substitute the coordinates xA and yA and we'll
get:

-3 = a*(-1) + b

-3 = -a +
b (1)

B belongs to the line y = ax+b if and only if yB =
a*xB + b.

We'll substitute the coordinates xB and yB and
we'll get:

7 = a*3 + b
(2)

We'll subtract (2) from
(1):

3a + b + a - b = 7 +
3

We'll combine and eliminate like
terms:

4a = 10

a =
5/2

b - a = -3

b = a -
3

b = 5/2 - 3

b =
-1/2

The function f(x) whose graph is represented by the
line AB:

f(x) = (5/2)*x -
1/2