Find the Parabola Through (0,63) with Vertex (5,-12) (5,-12) , (0,63)

$(5, - 12)$ , $(0, 63)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(5, - 12)$ as the vertex $(h, k)$ and $(0, 63)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$63 = a {(0 - (5))}^{2} - 12$

Step 2

Using $63 = a {(0 - (5))}^{2} - 12$ to solve for $a$ , $a = 3$ .

Rewrite the equation as $a {(0 - (5))}^{2} - 12 = 63$ .

$a {(0 - (5))}^{2} - 12 = 63$

Simplify $a {(0 - (5))}^{2} - 12$ .

Subtract $5$ from $0$ .

$a {(- (5))}^{2} - 12 = 63$

Simplify each term.

Multiply $- 1$ by $5$ .

$a {(- 5)}^{2} - 12 = 63$

Raise $- 5$ to the power of $2$ .

$a \cdot 25 - 12 = 63$

Move $25$ to the left of $a$ .

$25 a - 12 = 63$

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

Add $12$ to both sides of the equation.

$25 a = 63 + 12$

Add $63$ and $12$ .

$25 a = 75$

Divide each term in $25 a = 75$ by $25$ and simplify.

Divide each term in $25 a = 75$ by $25$ .

$\frac{25 a}{25} = \frac{75}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 a}{25} = \frac{75}{25}$

Divide $a$ by $1$ .

$a = \frac{75}{25}$

Simplify the right side.

Divide $75$ by $25$ .

$a = 3$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(5, - 12)$ and $a = 3$ is $y = (3) {(x - (5))}^{2} - 12$ .

$y = (3) {(x - (5))}^{2} - 12$

Step 4

Solve $y = (3) {(x - (5))}^{2} - 12$ for $y$ .

Remove parentheses.

$y = (3) {(x - (5))}^{2} - 12$

Multiply $3$ by ${(x - (5))}^{2}$ .

$y = 3 {(x - (5))}^{2} - 12$

Remove parentheses.

$y = (3) {(x - (5))}^{2} - 12$

Simplify $(3) {(x - (5))}^{2} - 12$ .

Simplify each term.

Multiply $- 1$ by $5$ .

$y = 3 {(x - 5)}^{2} - 12$

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$y = 3 ((x - 5) (x - 5)) - 12$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$y = 3 (x (x - 5) - 5 (x - 5)) - 12$

Apply the distributive property.

$y = 3 (x \cdot x + x \cdot - 5 - 5 (x - 5)) - 12$

Apply the distributive property.

$y = 3 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = 3 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) - 12$

Move $- 5$ to the left of $x$ .

$y = 3 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) - 12$

Multiply $- 5$ by $- 5$ .

$y = 3 (x^{2} - 5 x - 5 x + 25) - 12$

Subtract $5 x$ from $- 5 x$ .

$y = 3 (x^{2} - 10 x + 25) - 12$

Apply the distributive property.

$y = 3 x^{2} + 3 (- 10 x) + 3 \cdot 25 - 12$

Simplify.

Multiply $- 10$ by $3$ .

$y = 3 x^{2} - 30 x + 3 \cdot 25 - 12$

Multiply $3$ by $25$ .

$y = 3 x^{2} - 30 x + 75 - 12$

Subtract $12$ from $75$ .

$y = 3 x^{2} - 30 x + 63$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 3 x^{2} - 30 x + 63$

Vertex Form: $y = (3) {(x - (5))}^{2} - 12$

Step 6

Simplify the standard form.

Standard Form: $y = 3 x^{2} - 30 x + 63$

Vertex Form: $y = 3 {(x - 5)}^{2} - 12$

Step 7

Find the Parabola Through (0,63) with Vertex (5,-12) (5,-12) , (0,63)

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