Find the Parabola Through (5,-18) with Vertex (2,-9) (2,-9) , (5,-18)

$(2, - 9)$ , $(5, - 18)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(2, - 9)$ as the vertex $(h, k)$ and $(5, - 18)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 18 = a {(5 - (2))}^{2} - 9$

Step 2

Using $- 18 = a {(5 - (2))}^{2} - 9$ to solve for $a$ , $a = - 1$ .

Rewrite the equation as $a {(5 - (2))}^{2} - 9 = - 18$ .

$a {(5 - (2))}^{2} - 9 = - 18$

Simplify each term.

Multiply $- 1$ by $2$ .

$a {(5 - 2)}^{2} - 9 = - 18$

Subtract $2$ from $5$ .

$a \cdot 3^{2} - 9 = - 18$

Raise $3$ to the power of $2$ .

$a \cdot 9 - 9 = - 18$

Move $9$ to the left of $a$ .

$9 a - 9 = - 18$

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

Add $9$ to both sides of the equation.

$9 a = - 18 + 9$

Add $- 18$ and $9$ .

$9 a = - 9$

Divide each term in $9 a = - 9$ by $9$ and simplify.

Divide each term in $9 a = - 9$ by $9$ .

$\frac{9 a}{9} = \frac{- 9}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} = \frac{- 9}{9}$

Divide $a$ by $1$ .

$a = \frac{- 9}{9}$

Simplify the right side.

Divide $- 9$ by $9$ .

$a = - 1$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(2, - 9)$ and $a = - 1$ is $y = (- 1) {(x - (2))}^{2} - 9$ .

$y = (- 1) {(x - (2))}^{2} - 9$

Step 4

Solve $y = (- 1) {(x - (2))}^{2} - 9$ for $y$ .

Remove parentheses.

$y = (- 1) {(x - (2))}^{2} - 9$

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

$y = - 1 {(x - (2))}^{2} - 9$

Remove parentheses.

$y = (- 1) {(x - (2))}^{2} - 9$

Simplify $(- 1) {(x - (2))}^{2} - 9$ .

Simplify each term.

Multiply $- 1$ by $2$ .

$y = - 1 {(x - 2)}^{2} - 9$

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$y = - 1 ((x - 2) (x - 2)) - 9$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$y = - 1 (x (x - 2) - 2 (x - 2)) - 9$

Apply the distributive property.

$y = - 1 (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 9$

Apply the distributive property.

$y = - 1 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 9$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - 1 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) - 9$

Move $- 2$ to the left of $x$ .

$y = - 1 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) - 9$

Multiply $- 2$ by $- 2$ .

$y = - 1 (x^{2} - 2 x - 2 x + 4) - 9$

Subtract $2 x$ from $- 2 x$ .

$y = - 1 (x^{2} - 4 x + 4) - 9$

Apply the distributive property.

$y = - 1 x^{2} - 1 (- 4 x) - 1 \cdot 4 - 9$

Simplify.

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

$y = - x^{2} - 1 (- 4 x) - 1 \cdot 4 - 9$

Multiply $- 4$ by $- 1$ .

$y = - x^{2} + 4 x - 1 \cdot 4 - 9$

Multiply $- 1$ by $4$ .

$y = - x^{2} + 4 x - 4 - 9$

Subtract $9$ from $- 4$ .

$y = - x^{2} + 4 x - 13$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - x^{2} + 4 x - 13$

Vertex Form: $y = (- 1) {(x - (2))}^{2} - 9$

Step 6

Simplify the standard form.

Standard Form: $y = - x^{2} + 4 x - 13$

Vertex Form: $y = - 1 {(x - 2)}^{2} - 9$

Step 7

Find the Parabola Through (5,-18) with Vertex (2,-9) (2,-9) , (5,-18)

Related Questions

Support LCMGCF.com by disabling your adblocker.