Answer:
−
5
=
−
2
3
(
9
)
+
b
−
5
=
−
2
3
⋅
9
1
+
b
−
5
=
−
2
⋅
9
3
⋅
1
+
b
−
5
=
−
18
3
+
b
−
5
=
−
6
+
b
-6
Explanation:
Step 1: Pick two points from the table and plug these points into the rate of change equation. This equation is:
x y
-3 3
3 -1
9 -5
15 -9
21 -13
Δ
Y
Δ
X
=
Y
final
−
Y
intial
X
final
−
X
intial
. It is important that we start with the second value when we substitute in the
X
and
Y
values.
Step 2: Simplify the quotient to determine the average rate of change of the table.
Step 3: Replace
m
, in the linear function
y
=
m
x
+
b
, with the average rate of change you calculated in the previous step.
Step 4: Replace
x
and
y
in the
y
=
m
x
+
b
equation with any point from the given table.
Step 5: Simplify the right side of the equal sign.
Step 6: Use inverse operations to solve for
b
. This is the initial amount of the given table.
Rate of Change: This the ratio of the change in
y
to the change in
x
. It can be given by the equation:
Δ
Y
Δ
X
=
Y
final
−
Y
intial
X
final
−
X
initial
Initial Amount: This is the
y
value of a table that is produced when
x
is equal to 0.
So, let's try using these steps to find the initial amount and rate of change given a table for a linear function, in the following two examples!