Find the Parabola Through (0,-8) with Vertex (-1.5,-12.5) (-1.5,-12.5) , (0,-8)

$(- 1.5, - 12.5)$ , $(0, - 8)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 1.5, - 12.5)$ as the vertex $(h, k)$ and $(0, - 8)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 8 = a {(0 - (- 1.5))}^{2} - 12.5$

Step 2

Using $- 8 = a {(0 - (- 1.5))}^{2} - 12.5$ to solve for $a$ , $a = 2$ .

Rewrite the equation as $a {(0 - (- 1.5))}^{2} - 12.5 = - 8$ .

$a {(0 - (- 1.5))}^{2} - 12.5 = - 8$

Simplify $a {(0 - (- 1.5))}^{2} - 12.5$ .

Subtract $- 1.5$ from $0$ .

$a {(- (- 1.5))}^{2} - 12.5 = - 8$

Simplify each term.

Multiply $- 1$ by $- 1.5$ .

$a \cdot {1.5}^{2} - 12.5 = - 8$

Raise $1.5$ to the power of $2$ .

$a \cdot 2.25 - 12.5 = - 8$

Move $2.25$ to the left of $a$ .

$2.25 a - 12.5 = - 8$

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

Add $12.5$ to both sides of the equation.

$2.25 a = - 8 + 12.5$

Add $- 8$ and $12.5$ .

$2.25 a = 4.5$

Divide each term in $2.25 a = 4.5$ by $2.25$ and simplify.

Divide each term in $2.25 a = 4.5$ by $2.25$ .

$\frac{2.25 a}{2.25} = \frac{4.5}{2.25}$

Simplify the left side.

Cancel the common factor of $2.25$ .

Cancel the common factor.

$\frac{2.25 a}{2.25} = \frac{4.5}{2.25}$

Divide $a$ by $1$ .

$a = \frac{4.5}{2.25}$

Simplify the right side.

Divide $4.5$ by $2.25$ .

$a = 2$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 1.5, - 12.5)$ and $a = 2$ is $y = (2) {(x - (- 1.5))}^{2} - 12.5$ .

$y = (2) {(x - (- 1.5))}^{2} - 12.5$

Step 4

Solve $y = (2) {(x - (- 1.5))}^{2} - 12.5$ for $y$ .

Remove parentheses.

$y = (2) {(x - (- 1.5))}^{2} - 12.5$

Multiply $2$ by ${(x - (- 1.5))}^{2}$ .

$y = 2 {(x - (- 1.5))}^{2} - 12.5$

Remove parentheses.

$y = (2) {(x - (- 1.5))}^{2} - 12.5$

Simplify $(2) {(x - (- 1.5))}^{2} - 12.5$ .

Simplify each term.

Multiply $- 1$ by $- 1.5$ .

$y = 2 {(x + 1.5)}^{2} - 12.5$

Rewrite ${(x + 1.5)}^{2}$ as $(x + 1.5) (x + 1.5)$ .

$y = 2 ((x + 1.5) (x + 1.5)) - 12.5$

Expand $(x + 1.5) (x + 1.5)$ using the FOIL Method.

Apply the distributive property.

$y = 2 (x (x + 1.5) + 1.5 (x + 1.5)) - 12.5$

Apply the distributive property.

$y = 2 (x \cdot x + x \cdot 1.5 + 1.5 (x + 1.5)) - 12.5$

Apply the distributive property.

$y = 2 (x \cdot x + x \cdot 1.5 + 1.5 x + 1.5 \cdot 1.5) - 12.5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = 2 (x^{2} + x \cdot 1.5 + 1.5 x + 1.5 \cdot 1.5) - 12.5$

Move $1.5$ to the left of $x$ .

$y = 2 (x^{2} + 1.5 \cdot x + 1.5 x + 1.5 \cdot 1.5) - 12.5$

Multiply $1.5$ by $1.5$ .

$y = 2 (x^{2} + 1.5 x + 1.5 x + 2.25) - 12.5$

Add $1.5 x$ and $1.5 x$ .

$y = 2 (x^{2} + 3 x + 2.25) - 12.5$

Apply the distributive property.

$y = 2 x^{2} + 2 (3 x) + 2 \cdot 2.25 - 12.5$

Simplify.

Multiply $3$ by $2$ .

$y = 2 x^{2} + 6 x + 2 \cdot 2.25 - 12.5$

Multiply $2$ by $2.25$ .

$y = 2 x^{2} + 6 x + 4.5 - 12.5$

Subtract $12.5$ from $4.5$ .

$y = 2 x^{2} + 6 x - 8$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 2 x^{2} + 6 x - 8$

Vertex Form: $y = (2) {(x - (- 1.5))}^{2} - 12.5$

Step 6

Simplify the standard form.

Standard Form: $y = 2 x^{2} + 6 x - 8$

Vertex Form: $y = 2 {(x + 1.5)}^{2} - 12.5$

Step 7

Find the Parabola Through (0,-8) with Vertex (-1.5,-12.5) (-1.5,-12.5) , (0,-8)

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