Find the Parabola Through (1,2) with Vertex (0,0) (0,0) , (1,2)

$(0, 0)$ , $(1, 2)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, 0)$ as the vertex $(h, k)$ and $(1, 2)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$2 = a {(1 - (0))}^{2} + 0$

Step 2

Using $2 = a {(1 - (0))}^{2} + 0$ to solve for $a$ , $a = 2$ .

Rewrite the equation as $a {(1 - (0))}^{2} + 0 = 2$ .

$a {(1 - (0))}^{2} + 0 = 2$

Simplify $a {(1 - (0))}^{2} + 0$ .

Add $a {(1 - (0))}^{2}$ and $0$ .

$a {(1 - (0))}^{2} = 2$

Subtract $0$ from $1$ .

$a \cdot 1^{2} = 2$

One to any power is one.

$a \cdot 1 = 2$

Multiply $a$ by $1$ .

$a = 2$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, 0)$ and $a = 2$ is $y = (2) {(x - (0))}^{2} + 0$ .

$y = (2) {(x - (0))}^{2} + 0$

Step 4

Solve $y = (2) {(x - (0))}^{2} + 0$ for $y$ .

Remove parentheses.

$y = (2) {(x - (0))}^{2} + 0$

Multiply $2$ by ${(x - (0))}^{2}$ .

$y = 2 {(x - (0))}^{2} + 0$

Remove parentheses.

$y = (2) {(x - (0))}^{2} + 0$

Simplify $(2) {(x - (0))}^{2} + 0$ .

Add $(2) {(x - (0))}^{2}$ and $0$ .

$y = (2) {(x - (0))}^{2}$

Subtract $0$ from $x$ .

$y = 2 x^{2}$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 2 x^{2}$

Vertex Form: $y = (2) {(x - (0))}^{2} + 0$

Step 6

Simplify the standard form.

Standard Form: $y = 2 x^{2}$

Vertex Form: $y = 2 x^{2}$

Step 7

Find the Parabola Through (1,2) with Vertex (0,0) (0,0) , (1,2)

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