# What is the quadratic function of a graph if the points are (-2, -4) (-1, -2.5) (0, 0) (1, 3.5) (2, 8)?

Jun 4, 2015

It can be guessed since we deal with easy numbers, but let's approach this problem theoretically.

The general equation of a quadratic function looks like this:
$y = a {x}^{2} + b x + c$
There are three unknown coefficients here - $a$, $b$ and $c$

Let's use three points out of 5 given to determine these three coefficients.

Point $\left(- 2 , - 4\right)$ results in equation
$- 4 = a {\left(- 2\right)}^{2} + b \left(- 2\right) + c$
or, simplifying this,
(1) $- 4 = 4 a - 2 b + c$

Point $\left(- 1 , - 2.5\right)$ results in equation
$- 2.5 = a {\left(- 1\right)}^{2} + b \left(- 1\right) + c$
or, simplifying this,
(2) $- 2.5 = a - b + c$

Point $\left(0 , 0\right)$ results in equation
$0 = a {\left(0\right)}^{2} + b \left(0\right) + c$
or, simplifying this,
(3) $0 = c$

Equations (1), (2) and (3) constitute a system of 3 linear equations with three unknown variables. Let's solve it by substitution.

Step 1.
Substitute $c = 0$ from equation (3) into equations (1) and (2):
(1) $- 4 = 4 a - 2 b$
(2) $- 2.5 = a - b$

Step2.
Simplify the equation (1) by dividing left and right sides by 2:
(1) $- 2 = 2 a - b$
Solve it for $b$:
$b = 2 a + 2$

Step 3.
Substitute an expression for $b$ into equation (2):
(2) $- 2.5 = a - 2 a - 2$
or
$a = 0.5$

Step 4.
Find the value of $b$:
$b = 2 \cdot 0.5 + 2 = 3$

Together with previously determined $c = 0$, we have an equation:
$y = 0.5 {x}^{2} + 3 x$

All we have to do is to check that two other points specified in the problem lie on this graph.

Point $\left(1 , 3.5\right)$:
$0.5 \cdot {1}^{2} + 3 \cdot 1 = 0.5 + 3 = 3.5$ (check!)
Point $\left(2 , 8\right)$:
$0.5 \cdot {2}^{2} + 3 \cdot 2 = 0.5 \cdot 4 + 6 = 2 + 6 = 8$ (check!)

So, the answer to this problem is
$y = 0.5 {x}^{2} + 3 x$