15 points what is the equation of the following graph in vertex form? (parabolic function going down from the left through the point negative four comma zero and turning at the point negative three comma negative one and going up through the point negative two comma zero and then through the point zero comma eight and continuing up towards infinity) y = (x − 3)2 − 1 y = (x + 3)2 − 1 y = (x − 4)2 − 2 y = (x − 4)2 + 8

Answer with explanation:

→The given curve is Quadratic,having two real roots,as it cuts X axis at two distinct points that is at ,-2 and -4.

→Graph is symmetrical about line , x=-3.

→This curve is a Parabola,opening in upward direction ,that is having Y axis as Axis of Symmetry.

→Parabola is defined as locus of all the points such that distance from fixed point called focus, to distance from fixed line called Directrix ,is always Constant.

→Point which divides PARABOLA in two symmetrical parts is called Vertex having coordinates (-3,-1).

Equation of parabola having vertex ,(-3,-1) and opening in Upward Direction,that is in vertical direction, is given by:

General Equation: Equation of parabola passing through point (a,b) and opening in upward direction and having focus (k,h) is

y-b=4 k× (x-a)²

So, the  equation of parabola will be

→y-(-1)=4 k[x-(-3)]²

→y+1==4 k(x+3)²-----(1)

This parabola also passes through (-4,0) and (-2,0).

→0+1=4 k×(-4+3)²

→1=4 k×(-1)²

If you will substitute , x=-2 and y=0 in equation 1,you will get same value of k.

Equation of Parabola ,that is equation 1,can be written as

Option B: y=(x+3)²-1

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!