Answer:
[tex]a = -2[/tex]
Step-by-step explanation:
Given
The attached function (graph)
Required
Find a
The given function is quadratic and will be solved using:
[tex]y = ax^2 + bx + c[/tex]
From the attachment:
[tex](x,y) = (-3,4)[/tex]
[tex](x,y) = (-2,2)[/tex]
[tex](x,y) = (-4,2)[/tex]
Substitute these values in [tex]y = ax^2 + bx + c[/tex]
For: [tex](x,y) = (-3,4)[/tex]
[tex]4 = a(-3)^2 + b(-3) + c[/tex]
[tex]4 = 9a -3b + c[/tex]
For: [tex](x,y) = (-2,2)[/tex]
[tex]2 = a(-2)^2 + b(-2) + c[/tex]
[tex]2 = 4a -2b + c[/tex]
For: [tex](x,y) = (-4,2)[/tex]
[tex]2 = a(-4)^2 + b(-4) + c[/tex]
[tex]2 = 16a -4b + c[/tex]
So, we have:
[tex]4 = 9a -3b + c[/tex] --- (1)
[tex]2 = 4a -2b + c[/tex] ---- (2)
[tex]2 = 16a -4b + c[/tex] --- (3)
Make c the subject in (1)
[tex]c = 4 - 9a + 3b[/tex]
Substitute [tex]c = 4 - 9a + 3b[/tex] in (2) and (3)
[tex]2 = 4a -2b + c[/tex]
[tex]2 = 4a -2b + 4 - 9a + 3b[/tex]
Collect Like Terms
[tex]-4 + 2 = 4a - 9a-2b + 3b[/tex]
[tex]-2 = - 5a+b[/tex] --- (4)
[tex]2 = 16a -4b + c[/tex]
[tex]2 = 16a - 4b + 4 - 9a + 3b[/tex]
Collect Like Terms
[tex]2 -4 = 16a - 9a + 3b- 4b[/tex]
[tex]-2 = 7a -b[/tex] --- (5)
Add 4 and 5
[tex]-2 = - 5a+b[/tex]
[tex]-2 = 7a -b[/tex]
--------------------------
[tex]-2 - 2 = -5a + 7a +b - b[/tex]
[tex]-4 = 2a[/tex]
Divide through by 2
[tex]-2 = a[/tex]
[tex]a = -2[/tex]
