Determine the quadratic polynomial in one variable whose graph passes thru the points (1,6), (2,3) and (3,2).
let $\displaystyle f(x)$ be a curve. a point $\displaystyle (a,b)$ is on the curve if $\displaystyle f(a)=b$.
if you want a quadratic in one variable, say in $\displaystyle x$, then your it is of the form $\displaystyle f(x)=ax^2 + bx + c$
you are given 3 points. then you can solve $\displaystyle a,b,c$ by solving the system of equation
$\displaystyle f(1)=6$
$\displaystyle f(2)=3$
$\displaystyle f(3)=2$
Q)Determine the quadratic polynomial in one variable whose graph passes thru the points (1,6), (2,3) and (3,2).
ANS)Cont.
Thus we have
6 = a + b + c --------(1)
3 = 4a + 2b + c --------(2)
2 = 9a + 3b + c --------(3)
Multiply (1) by 2 and subtract from (2)
Multiply (1) by 3 and subtract from (3)
We get
2a-c= -9 à c-2a=9 --------(4)
And
6a-2c = -16 à 3a-c=-8 --------(5)
Adding (4) and (5) We get
a=1
Putting a in (4) We get
c-2(1)=9 à c-2=9 à c=11
putting a and c in (1)
6=1+b+11 à 6=12+b à b=-6
Hence in order we get
a=1,b=-6,c=11 --------(6)
We have
F(x) =ax2 + bx + c --------(7)
Putting values of f and (6) in (7) we get
x2 - 6x + 5 = 0
x2 - 6x + 8 = 0
x2 - 6x + 9 = 0
Is This right I am getting three equations whats wrong.
Hello, puneet!
With this type of system of equations,
. . there is a much simpler approach.
. . $\displaystyle \begin{array}{cccc}a + b + c &=& 6 & [1] \\
4a + 2b + c &=& 3 & [2] \\
9a + 3b + c &=& 2 & [3] \end{array}$
Eliminate $\displaystyle c.$
. . $\displaystyle \begin{array}{ccccc}\text{Subtract: [2] - [1]} & 3a + b &=& \text{-}3 & [4] \\
\text{Subtract: [3] - [2]} & 5a + b &=& \text{-}1 & [5] \end{array}$
Eliminate $\displaystyle b.$
. . $\displaystyle \text{Subtract: [5] - [4] }\;\;2a\:=\: 2 \quad\Rightarrow\quad \boxed{a \:=\:1}$
$\displaystyle \text{Substitute into [4]: }\;3 + b \:=\:\text{-}3 \quad\Rightarrow\quad\boxed{ b \:=\:\text{-}6}$
$\displaystyle \text{Substitute into [1]: }\;1 - 6 + c \:=\:6 \quad\Rightarrow\quad\boxed{ c \:=\:11}$
$\displaystyle \text{Therefore: }\;f(x) \;=\;x^2-6x + 11$