Find the Parabola Through (0,7) with Vertex (3,-2) (3,-2) , (0,7)

$(3, - 2)$ , $(0, 7)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(3, - 2)$ as the vertex $(h, k)$ and $(0, 7)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$7 = a {(0 - (3))}^{2} - 2$

Step 2

Using $7 = a {(0 - (3))}^{2} - 2$ to solve for $a$ , $a = 1$ .

Rewrite the equation as $a {(0 - (3))}^{2} - 2 = 7$ .

$a {(0 - (3))}^{2} - 2 = 7$

Simplify $a {(0 - (3))}^{2} - 2$ .

Subtract $3$ from $0$ .

$a {(- (3))}^{2} - 2 = 7$

Simplify each term.

Multiply $- 1$ by $3$ .

$a {(- 3)}^{2} - 2 = 7$

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

$a \cdot 9 - 2 = 7$

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

$9 a - 2 = 7$

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

Add $2$ to both sides of the equation.

$9 a = 7 + 2$

Add $7$ and $2$ .

$9 a = 9$

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 $(3, - 2)$ and $a = 1$ is $y = (1) {(x - (3))}^{2} - 2$ .

$y = (1) {(x - (3))}^{2} - 2$

Step 4

Solve $y = (1) {(x - (3))}^{2} - 2$ for $y$ .

Remove parentheses.

$y = (1) {(x - (3))}^{2} - 2$

Multiply $1$ by ${(x - (3))}^{2}$ .

$y = 1 {(x - (3))}^{2} - 2$

Remove parentheses.

$y = (1) {(x - (3))}^{2} - 2$

Simplify $(1) {(x - (3))}^{2} - 2$ .

Simplify each term.

Multiply ${(x - (3))}^{2}$ by $1$ .

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

Multiply $- 1$ by $3$ .

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

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y = (x - 3) (x - 3) - 2$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 3) - 3 (x - 3) - 2$

Apply the distributive property.

$y = x \cdot x + x \cdot - 3 - 3 (x - 3) - 2$

Apply the distributive property.

$y = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

$y = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 2$

Multiply $- 3$ by $- 3$ .

$y = x^{2} - 3 x - 3 x + 9 - 2$

Subtract $3 x$ from $- 3 x$ .

$y = x^{2} - 6 x + 9 - 2$

Subtract $2$ from $9$ .

$y = x^{2} - 6 x + 7$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = x^{2} - 6 x + 7$

Vertex Form: $y = (1) {(x - (3))}^{2} - 2$

Step 6

Simplify the standard form.

Standard Form: $y = x^{2} - 6 x + 7$

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

Step 7

Find the Parabola Through (0,7) with Vertex (3,-2) (3,-2) , (0,7)

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