we know that
A linear function can be modeled with the equation of a line
We have
case 1
A(-6,26) B(0,3) C(6,26) D(12,101)
Step 1
with point A and B find the equation of a line
Slope m=(y2-y1)/(x2-x1)------> m=(3-26)/(0+6)-----> m=-23/6
with m and point B find the equation of a line
y-y1=m*(x-x1)------>y-3=(-23/6)*x-----> y=(-23/6)x+3
Step 2
check if point C and point D belong to the line
C(6,26) D(12,101)
point C
for x=6 the value of y must be 26
y=(-23/6)*6+3-----> y=-20
-20 is not equals to 26
so
the ordered pairs of case 1 not represents a linear function
case 2
A(-6,-3) B(0,3) C(6,1) D(12,2)
Step 1
with point A and B find the equation of a line
Slope m=(y2-y1)/(x2-x1)------> m=(3+3)/(0+6)-----> m=1
with m and point B find the equation of a line
y-y1=m*(x-x1)------>y-3=(1)*x-----> y=x+3
Step 2
check if point C and point D belong to the line
C(6,1) D(12,2)
point C
for x=6 the value of y must be 1
y=6+3-----> y=9
9 is not equals to 1
so
the ordered pairs of case 2 not represents a linear function
case 3
A(-6,-3) B(0,3) C(6,9) D(6,3)
Step 1
with point A and B find the equation of a line
Slope m=(y2-y1)/(x2-x1)------> m=(3+3)/(0+6)-----> m=1
with m and point B find the equation of a line
y-y1=m*(x-x1)------>y-3=(1)*x-----> y=x+3
Step 2
check if point C and point D belong to the line
C(6,9) D(6,3)
point C
for x=6 the value of y must be 9
y=6+3-----> y=9
9 is equals to 1
point D
for x=6 the value of y must be 3
y=6+3-----> y=9
9 is not equals to 3
so
the ordered pairs of case 3 not represents a linear function
case 4
A(-6,-3) B(0,2) C(6,7) D(12,12)
Step 1
with point A and B find the equation of a line
Slope m=(y2-y1)/(x2-x1)------> m=(2+3)/(0+6)-----> m=5/6
with m and point B find the equation of a line
y-y1=m*(x-x1)------>y-2=(5/6)*x-----> y=(5/6)x+2
Step 2
check if point C and point D belong to the line
C(6,7) D(12,12)
point C
for x=6 the value of y must be 7
y=(5/6)*6+2-----> y=7
7 is equals to 7------> is ok
point D
for x=12 the value of y must be 12
y=(5/6)*12+2-----> y=12
12 is equals to 12------> is ok
so
the ordered pairs of case 4 represents a linear function
therefore
the answer is
the table xy −6−3 0 2 6 7 12 12 represents a linear function
