Question

The vertex of this parabola is at (4,-3). When the x-value is 5, the y-value is -6.What is the coefficient of the squared expression in the parabola's equation?

Answer 1

The coefficient of the squared expression in the parabola's equation is -3.

Step-by-step explanation:

We can solve this problem by using the vertex form of the parabola equation:

y = a(x - h)^2 + k

where:

a is the coefficient of the squared term

h is the x-coordinate of the vertex

k is the y-coordinate of the vertex

We are given that:

h = 4 (from the vertex coordinates)

k = -3 (from the vertex coordinates)

Now, we can use the additional information that the parabola passes through the point (5, -6). Substituting these values into the equation, we get:

-6 = a(5 - 4)^2 - 3

Solving for a:

-6 = a(1)^2 - 3

-6 = a - 3

-3 = a

Therefore, the coefficient of the squared term, a, is -3.

Answer 2

`-3x^(2)+24x-51`

Step-by-step explanation:

we have that the form of the parabola's equation is

`ax^(2)+bx+c`

an also we have

`a(4)^(2)+b(4)+c=-3\\a(5)^(2)+b(5)+c=-6`

and for the vertex in (4,-3)

`x=-(b)/(2a)`

`4=-(b)/(2a)\\b=-8a` (1)

if we subtract the first equation to the second equation we can obtain a 2x2 system equation

`9a+b=-3\\b=-3-9a` (2)

and by taking the equations (1) and (2)

`-8a=-3-9a\\a=-3`

hence, for b we have

`b=-8(a)=-8(-3)=24`

and to compute c we can use

`a(4)^(2)+b(4)+c=-3\\(-3)(16)+(24)(4)+c=-3\\c=-51`

Finally we have that the parabola is

`-3x^(2)+24x-51`

I hope this is useful for you

regards