Thread: How to find the equation of a Parabola

1. How to find the equation of a Parabola

Hi,

Could someone explain step-by-step, how to work through this type of sums:

"Find the equation of the quadratic which passes through the points with coordinates:

(-2,-1), (1,2), (3,-16)"

I substituted the values in to the equation $\displaystyle ax^2+bx+c$

$\displaystyle -1=4a-2b+c$
$\displaystyle 2 =a+b+c$
$\displaystyle -16=9a+3b+c$

I'm not sure how you work out the simultaneous equations though, Any help would be much appreciated,

Thanks,

Dru

2. Hello, 22upon7!

You did the hard part . . . and you forgot your algrbra?

Find the equation of the parabola which passes through the points
. . with coordinates: .(-2,-1), (1,2), (3,-16)

I substituted the values in to the equation $\displaystyle y \:=\:ax^2+bx+c$

$\displaystyle -1=4a-2b+c$
$\displaystyle 2 =a+b+c$
$\displaystyle -16=9a+3b+c$

We have: .$\displaystyle \begin{array}{cccc}a + b + c &=& 2 & {\color{blue}[1]} \\ 4a - 2b + c &=& \text{-}1 & {\color{blue}[2]}\\ 9a + 3b + c &=& \text{-}16 & {\color{blue}[3]} \end{array}$

Subtract $\displaystyle {\color{blue}[2] - [1]}\!:\;\;3a - 3b \:=\:\text{-}3 \;\;\quad\Rightarrow\quad a - b \:=\:\text{-}1\;\;{\color{blue}[4]}$

Subtract $\displaystyle {\color{blue}[3] - [2]}\!:\;\;5a + 5b \:=\:\text{-}15 \quad\Rightarrow\quad a + b \:=\:\text{-}3\;\;{\color{blue}[5]}$

Add $\displaystyle {\color{blue}[4] + [5]}\!:\;\;2a \:=\:-4 \quad\Rightarrow\quad\boxed{ a \:=\:-2}$

Substitute into $\displaystyle {\color{blue}[5]}\!:\;\;-2 + b \:=\:-3 \quad\Rightarrow\quad \boxed{b\:=\:-1}$

Substitute into $\displaystyle {\color{blue}[1]}\!:\;\;-2 -1 + c \:=\:2 \quad\Rightarrow\quad\boxed{ c \:=\:5}$

Therefore: .$\displaystyle y \;=\;-2x^2 - x + 5$

3. Originally Posted by Soroban
Hello, 22upon7!

You did the hard part . . . and you forgot your algrbra?

We have: .$\displaystyle \begin{array}{cccc}a + b + c &=& 2 & {\color{blue}[1]} \\ 4a - 2b + c &=& \text{-}1 & {\color{blue}[2]}\\ 9a + 3b + c &=& \text{-}16 & {\color{blue}[3]} \end{array}$

Subtract $\displaystyle {\color{blue}[2] - [1]}\!:\;\;3a - 3b \:=\:\text{-}3 \;\;\quad\Rightarrow\quad a - b \:=\:\text{-}1\;\;{\color{blue}[4]}$

Subtract $\displaystyle {\color{blue}[3] - [2]}\!:\;\;5a + 5b \:=\:\text{-}15 \quad\Rightarrow\quad a + b \:=\:\text{-}3\;\;{\color{blue}[5]}$

Add $\displaystyle {\color{blue}[4] + [5]}\!:\;\;2a \:=\:-4 \quad\Rightarrow\quad\boxed{ a \:=\:-2}$

Substitute into $\displaystyle {\color{blue}[5]}\!:\;\;-2 + b \:=\:-3 \quad\Rightarrow\quad \boxed{b\:=\:-1}$

Substitute into $\displaystyle {\color{blue}[1]}\!:\;\;-2 -1 + c \:=\:2 \quad\Rightarrow\quad\boxed{ c \:=\:5}$

Therefore: .$\displaystyle y \;=\;-2x^2 - x + 5$

Thankyou!