Answer:
Figure 1 shows a sketch of a curve C with equation y = f(x) and a straight line l.
Curve C meets l at the points (-2, 13) and (0,25) as shown.
The shaded region R is bounded by C and l as shown in Figure 1.
Given that
- f(x) is a quadratic function in x
- (-2, 13) is the minimum turning point of y = f(x)
Let's use inequalities to define R.
Step 1. Let's calculate the equation of the quadratic function.
As the vertex has coordinates
(
−
2
,
13
)
then
x
v
=
−
b
2
a
⟹
−
2
=
−
b
2
a
⟹
b
=
4
a
⋯
(
1
)
.
On the other hand, the equation of a quadratic function is given by
f
(
x
)
=
a
x
2
+
b
x
+
c
.
As the point
(
0
,
25
)
belongs to the function we have
25
=
c
.
Since the point
(
−
2
,
13
)
also belongs to the function we have then
13
=
a
(
−
2
)
2
+
b
(
−
2
)
+
25
13
=
4
a
−
2
b
+
25
⋯
(
2
)
.
Replacing
(
1
)
in
(
2
)
we have
13
=
4
a
−
2
(
4
a
)
+
25
13
=
−
4
a
+
25
⟹
a
=
3.
Replacing
a
=
3
in
(
1
)
we have
b
=
12.
Thus the equation of the quadratic function is
f
(
x
)
=
3
x
2
+
12
x
+
25.
Step 2. Calculation of the straight line.
Let be the points
(
−
2
,
13
)
and
(
0
,
25
)
.
Let's calculate the slope.
m
=
25
−
13
0
+
2
=
12
2
=
6.
Now we apply the equation point-slope
y
−
y
0
=
m
(
x
−
x
0
)
y
−
25
=
6
x
⟹
y
=
6
x
+
25.
Therefore R is the region bounded by the curves
f
(
x
)
=
3
x
2
+
12
x
+
25
and
g
(
x
)
=
6
x
+
25
.
Step-by-step explanation:
