Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Subtract $0$ from $1$ .

$a \cdot 1^{2} + 1 = 0$

Step 2.2.2

One to any power is one.

$a \cdot 1 + 1 = 0$

Step 2.2.3

Multiply $a$ by $1$ .

$a + 1 = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, 1)$ and $a = - 1$ is $y = (- 1) {(x - (0))}^{2} + 1$ .

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

Step 4

Solve $y = (- 1) {(x - (0))}^{2} + 1$ for $y$ .

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Step 4.1

Remove parentheses.

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

Step 4.2

Multiply $- 1$ by ${(x - (0))}^{2}$ .

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

Step 4.3

Remove parentheses.

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

Step 4.4

Simplify each term.

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Step 4.4.1

Subtract $0$ from $x$ .

$y = - 1 x^{2} + 1$

Step 4.4.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$y = - x^{2} + 1$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - x^{2} + 1$

Vertex Form: $y = (- 1) {(x - (0))}^{2} + 1$

Step 6

Simplify the standard form.

Standard Form: $y = - x^{2} + 1$

Vertex Form: $y = - 1 x^{2} + 1$

Step 7

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