quadratic rules | Racquel Sanderson

Tag Archives: quadratic rules

May 26, 2024 · 1:07 pm

Quadratic Rule from a Table of Values

How do you find a quadratic rule from a table of values?

For example,

$x$	1	2	3	4
$y$	0	6	14	24

Find the first difference

First Difference

6-0=6

14-6=8

24-14=10

Find the second difference (if the second difference is a constant, then it is quadratic)

Second Difference

8-6=2

10-8=2

The general equation of a quadratic is $y=ax^2+bx+c$

The second difference is $2a$

Hence our equation is now $y=x^2+bx+c$

The $c$ value is the vertical intercept ( $x=0$ ). We can back track in the table

$x$	0	1	2	3	4
$y$	$c$	0	6	14	24

As the first differences are 6, 8, 10, the one between 0 and 1 must be 4

$0-c=4$

$c=-4$

Our equation is now $y=x^2+bx-4$ .

We can now use any other point to find the $b$ value.

Let’s use the point $(2, 6)$

$6=2^2+b(2)-4$

$6=4-2b-4$

$6=2b$

$b=3$

The function is $y=x^2+3x-4$

Let’s try another one

$x$	3	4	5	6
$y$	7	17	31	49

First differences

First difference

Second difference

Second Difference

Hence $2a=4$ , therefore $a=2$

The equation is now $y=2x^2+bx+c$

Instead of back tracking, this time I am going to use two points and simultaneous equations.

Using points $(3, 7)$ and $(4, 17)$

$7=2(3)^2+b(3)+c$

(1) $\begin{equation*}3b+c=-11\end{equation*}$

$17=2(4)^2+b(4)+c$

(2) $\begin{equation*}4b+c=-15\end{equation*}$

Equation 2 – Equation 1

$b=-4$

Substitute $b=-4$ into equation 1

$3(-4)+c=-11$

$-12+c=-11$

$c=1$

Hence the equation is $y=2x^2-4x+1$

Tag Archives: quadratic rules

Quadratic Rule from a Table of Values

Recent Posts

Categories

Archives