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

$(0, 0)$ , $(2, - 2)$

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

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

Step 2

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

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

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

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

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

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

Subtract $0$ from $2$ .

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

Raise $2$ to the power of $2$ .

$a \cdot 4 = - 2$

Move $4$ to the left of $a$ .

$4 a = - 2$

Divide each term in $4 a = - 2$ by $4$ and simplify.

Divide each term in $4 a = - 2$ by $4$ .

$\frac{4 a}{4} = \frac{- 2}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{- 2}{4}$

Divide $a$ by $1$ .

$a = \frac{- 2}{4}$

Simplify the right side.

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

$a = \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$a = \frac{2 \cdot - 1}{2 \cdot 2}$

Cancel the common factor.

$a = \frac{2 \cdot - 1}{2 \cdot 2}$

Rewrite the expression.

$a = \frac{- 1}{2}$

Move the negative in front of the fraction.

$a = - \frac{1}{2}$

Step 3

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

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

Step 4

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

Remove parentheses.

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

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

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

Remove parentheses.

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

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

Simplify by adding zeros.

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

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

Subtract $0$ from $x$ .

$y = - \frac{1}{2} x^{2}$

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2}$

Step 5

The standard form and vertex form are as follows.

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

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

Step 6

Simplify the standard form.

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

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

Step 7

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

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