Precalculus Examples

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

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 $(1, 1)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Subtract $0$ from $1$ .

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

Step 2.2.3

One to any power is one.

$a \cdot 1 = 1$

Step 2.2.4

Multiply $a$ by $1$ .

$a = 1$

Step 3

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

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

Step 4

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

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Step 4.1

Remove parentheses.

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

Step 4.2

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

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

Step 4.3

Remove parentheses.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

Subtract $0$ from $x$ .

$y = x^{2}$

Step 5

The standard form and vertex form are as follows.

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

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

Step 6

Simplify the standard form.

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

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

Step 7

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