Find the Parabola Through (6,-5) with Vertex (0,175) (6,-5) , (0,175)

$(6, - 5)$ , $(0, 175)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, 175)$ as the vertex $(h, k)$ and $(6, - 5)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 5 = a {(6 - (0))}^{2} + 175$

Step 2

Using $- 5 = a {(6 - (0))}^{2} + 175$ to solve for $a$ , $a = - 5$ .

Rewrite the equation as $a {(6 - (0))}^{2} + 175 = - 5$ .

$a {(6 - (0))}^{2} + 175 = - 5$

Simplify each term.

Subtract $0$ from $6$ .

$a \cdot 6^{2} + 175 = - 5$

Raise $6$ to the power of $2$ .

$a \cdot 36 + 175 = - 5$

Move $36$ to the left of $a$ .

$36 a + 175 = - 5$

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

Subtract $175$ from both sides of the equation.

$36 a = - 5 - 175$

Subtract $175$ from $- 5$ .

$36 a = - 180$

Divide each term in $36 a = - 180$ by $36$ and simplify.

Divide each term in $36 a = - 180$ by $36$ .

$\frac{36 a}{36} = \frac{- 180}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 a}{36} = \frac{- 180}{36}$

Divide $a$ by $1$ .

$a = \frac{- 180}{36}$

Simplify the right side.

Divide $- 180$ by $36$ .

$a = - 5$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, 175)$ and $a = - 5$ is $y = (- 5) {(x - (0))}^{2} + 175$ .

$y = (- 5) {(x - (0))}^{2} + 175$

Step 4

Solve $y = (- 5) {(x - (0))}^{2} + 175$ for $y$ .

Remove parentheses.

$y = (- 5) {(x - (0))}^{2} + 175$

Multiply $- 5$ by ${(x - (0))}^{2}$ .

$y = - 5 {(x - (0))}^{2} + 175$

Remove parentheses.

$y = (- 5) {(x - (0))}^{2} + 175$

Subtract $0$ from $x$ .

$y = - 5 x^{2} + 175$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 5 x^{2} + 175$

Vertex Form: $y = (- 5) {(x - (0))}^{2} + 175$

Step 6

Simplify the standard form.

Standard Form: $y = - 5 x^{2} + 175$

Vertex Form: $y = - 5 x^{2} + 175$

Step 7

Find the Parabola Through (6,-5) with Vertex (0,175) (6,-5) , (0,175)

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