Passes through (-3,7) and has vertex (-5,4)

y

=

−

3

x

2

+

30

x

−

71

Explanation:

the equation of a parabola in  

vertex form

is.

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

∣

∣

∣

2

2

y

=

a

(

x

−

h

)

2

+

k

2

2

∣

∣

∣

−−−−−−−−−−−−−−−−−−−−−  

where  

(

h

,

k

)

are the coordinates of the vertex and a

is a multiplier

here  

(

h

,

k

)

=

(

5

,

4

)

⇒

y

=

a

(

x

−

5

)

2

+

4

to find a substitute  

(

7

,

−

8

)

into the equation

−

8

=

4

a

+

4

⇒

a

=

−

3

⇒

y

=

−

3

(

x

−

5

)

2

+

4

←

in vertex form

distributing and simplifying gives

y

=

−

3

(

x

2

−

10

x

+

25

)

+

4

y

=

−

3

x

2

+

30

x

−

75

+

4

⇒

y

=

−

3

x

2

+

30

x

−

71

←

in standard form

Step-by-step explanation:

step-by-step explanation:

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4