Question

Select the correct answer. What is the value of a in the function’s equation?

Answer 1

`a = -2`

Explanation:

Given

The attached function (graph)

Required

Find a

The given function is quadratic and will be solved using:

`y = ax^2 + bx + c`

From the attachment:

`(x,y) = (-3,4)`

`(x,y) = (-2,2)`

`(x,y) = (-4,2)`

Substitute these values in
`y = ax^2 + bx + c`

For:
`(x,y) = (-3,4)`

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

`4 = 9a -3b + c`

For:
`(x,y) = (-2,2)`

`2 = a(-2)^2 + b(-2) + c`

`2 = 4a -2b + c`

For:
`(x,y) = (-4,2)`

`2 = a(-4)^2 + b(-4) + c`

`2 = 16a -4b + c`

So, we have:

`4 = 9a -3b + c` --- (1)

`2 = 4a -2b + c` ---- (2)

`2 = 16a -4b + c` --- (3)

Make c the subject in (1)

`c = 4 - 9a + 3b`

Substitute
`c = 4 - 9a + 3b` in (2) and (3)

`2 = 4a -2b + c`

`2 = 4a -2b + 4 - 9a + 3b`

Collect Like Terms

`-4 + 2 = 4a - 9a-2b + 3b`

`-2 = - 5a+b` --- (4)

`2 = 16a -4b + c`

`2 = 16a - 4b + 4 - 9a + 3b`

Collect Like Terms

`2 -4 = 16a - 9a + 3b- 4b`

`-2 = 7a -b` --- (5)

Add 4 and 5

`-2 = - 5a+b`

`-2 = 7a -b`

--------------------------

`-2 - 2 = -5a + 7a +b - b`

`-4 = 2a`

Divide through by 2

`-2 = a`

`a = -2`