Find the Parabola Through (4,-1) with Vertex (0,15) (4,-1) , (0,15)

$(4, - 1)$ , $(0, 15)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, 15)$ as the vertex $(h, k)$ and $(4, - 1)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 1 = a {(4 - (0))}^{2} + 15$

Step 2

Using $- 1 = a {(4 - (0))}^{2} + 15$ to solve for $a$ , $a = - 1$ .

Rewrite the equation as $a {(4 - (0))}^{2} + 15 = - 1$ .

$a {(4 - (0))}^{2} + 15 = - 1$

Simplify each term.

Subtract $0$ from $4$ .

$a \cdot 4^{2} + 15 = - 1$

Raise $4$ to the power of $2$ .

$a \cdot 16 + 15 = - 1$

Move $16$ to the left of $a$ .

$16 a + 15 = - 1$

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

Subtract $15$ from both sides of the equation.

$16 a = - 1 - 15$

Subtract $15$ from $- 1$ .

$16 a = - 16$

Divide each term in $16 a = - 16$ by $16$ and simplify.

Divide each term in $16 a = - 16$ by $16$ .

$\frac{16 a}{16} = \frac{- 16}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 a}{16} = \frac{- 16}{16}$

Divide $a$ by $1$ .

$a = \frac{- 16}{16}$

Simplify the right side.

Divide $- 16$ by $16$ .

$a = - 1$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, 15)$ and $a = - 1$ is $y = (- 1) {(x - (0))}^{2} + 15$ .

$y = (- 1) {(x - (0))}^{2} + 15$

Step 4

Solve $y = (- 1) {(x - (0))}^{2} + 15$ for $y$ .

Remove parentheses.

$y = (- 1) {(x - (0))}^{2} + 15$

Multiply $- 1$ by ${(x - (0))}^{2}$ .

$y = - 1 {(x - (0))}^{2} + 15$

Remove parentheses.

$y = (- 1) {(x - (0))}^{2} + 15$

Simplify each term.

Subtract $0$ from $x$ .

$y = - 1 x^{2} + 15$

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

$y = - x^{2} + 15$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - x^{2} + 15$

Vertex Form: $y = (- 1) {(x - (0))}^{2} + 15$

Step 6

Simplify the standard form.

Standard Form: $y = - x^{2} + 15$

Vertex Form: $y = - 1 x^{2} + 15$

Step 7

Find the Parabola Through (4,-1) with Vertex (0,15) (4,-1) , (0,15)

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