# What is the equation, in standard form, of a parabola that contains the following points (–2, 18), (0, 4), (4, 42)?

Apr 21, 2018

$y = \frac{11}{4} {x}^{2} - \frac{3}{2} x + 4$

#### Explanation:

The standard form of a quadratic equation is

$y = a {x}^{2} + b x + c$.

Here, when $x = 0$, $y = 4$, so it is easy to determine $c$.

$4 = a \cdot {0}^{2} + b \cdot 0 + c = c$

$c = 4$

We have two more points with which to create two equations and two unknowns, $a$, and $b$.

$18 = a {\left(- 2\right)}^{2} + b \left(- 2\right) + 4 = 4 a - 2 b + 4$

$42 = a {\left(4\right)}^{2} + b \left(4\right) + 4 = 16 a + 4 b + 4$

We can subtract 4 from both sides of these equations and divide the second equation by 2.

$4 a - 2 b = 14$

$8 a + 2 b = 19$

Now add the two equations together.

$12 a = 33$

$a = \frac{11}{4}$

Substitute this value of $a$ into either equation to get

$b = - \frac{3}{2}$

This makes our final equation

$y = \frac{11}{4} {x}^{2} - \frac{3}{2} x + 4$