Solve y = -7(-13)

I'm giving 30 points!

y = -7(-13)

=> y = -7 × (-13)

= y = 91

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:

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