Find the Parabola Through (1,-9) with Vertex (0,-6) (1,-9) , (0,-6)

$(1, - 9)$ , $(0, - 6)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, - 6)$ as the vertex $(h, k)$ and $(1, - 9)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 9 = a {(1 - (0))}^{2} - 6$

Step 2

Using $- 9 = a {(1 - (0))}^{2} - 6$ to solve for $a$ , $a = - 3$ .

Rewrite the equation as $a {(1 - (0))}^{2} - 6 = - 9$ .

$a {(1 - (0))}^{2} - 6 = - 9$

Simplify each term.

Subtract $0$ from $1$ .

$a \cdot 1^{2} - 6 = - 9$

One to any power is one.

$a \cdot 1 - 6 = - 9$

Multiply $a$ by $1$ .

$a - 6 = - 9$

Move all terms not containing $a$ to the right side of the equation.

Add $6$ to both sides of the equation.

$a = - 9 + 6$

Add $- 9$ and $6$ .

$a = - 3$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, - 6)$ and $a = - 3$ is $y = (- 3) {(x - (0))}^{2} - 6$ .

$y = (- 3) {(x - (0))}^{2} - 6$

Step 4

Solve $y = (- 3) {(x - (0))}^{2} - 6$ for $y$ .

Remove parentheses.

$y = (- 3) {(x - (0))}^{2} - 6$

Multiply $- 3$ by ${(x - (0))}^{2}$ .

$y = - 3 {(x - (0))}^{2} - 6$

Remove parentheses.

$y = (- 3) {(x - (0))}^{2} - 6$

Subtract $0$ from $x$ .

$y = - 3 x^{2} - 6$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 3 x^{2} - 6$

Vertex Form: $y = (- 3) {(x - (0))}^{2} - 6$

Step 6

Simplify the standard form.

Standard Form: $y = - 3 x^{2} - 6$

Vertex Form: $y = - 3 x^{2} - 6$

Step 7

Find the Parabola Through (1,-9) with Vertex (0,-6) (1,-9) , (0,-6)

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