Question

Use a system of linear equations to find the quadratic function

Answer 1

Answer:

$f(x)=-2x^2-3x-9$

Step-by-step explanation:

Let $f(x)=ax^2+bx+c$ be the quadratic function.

$f(-2)=-11$ implies that $a(-2)^2+b(-2)+c=-11$

$4a-2b+c=-11......(1)$

Similarly, $f(1)=-5\implies a(1)^2+b(1)+c=-5\implies a+b=-5....(2)$

and $f(2)=-23\implies a(2)^2+b(2)+c=-23\implies 4a+2b+c=-23...(3)$

From equation 2, $a=-b-5...(4)$

Put (4) into (1) to get:

$4(-b-5)-2b+c=-11\implies -6b+c=9....(5)$

Put (4) in (3) to get:

$4(-b-5)+2b+c=-23\implies -2b+c=-3---(6)$

Subtract (6) from (5) to get:

$-4b=12$

$b=-3$

Put b=-3 in (4) to get:

a=--3-5=-2

Put b=-3 in to (6) to get:

-2(-3)+c=-3

6+c=-3

c=-6+-3=-9

Therefore the required equation is $f(x)=-2x^2-3x-9$