Find the Parabola Through (-5,80) with Vertex (0,-45) (-5,80) , (0,-45)

$(- 5, 80)$ , $(0, - 45)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, - 45)$ as the vertex $(h, k)$ and $(- 5, 80)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$80 = a {(- 5 - (0))}^{2} - 45$

Step 2

Using $80 = a {(- 5 - (0))}^{2} - 45$ to solve for $a$ , $a = 5$ .

Rewrite the equation as $a {(- 5 - (0))}^{2} - 45 = 80$ .

$a {(- 5 - (0))}^{2} - 45 = 80$

Simplify each term.

Subtract $0$ from $- 5$ .

$a {(- 5)}^{2} - 45 = 80$

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

$a \cdot 25 - 45 = 80$

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

$25 a - 45 = 80$

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

Add $45$ to both sides of the equation.

$25 a = 80 + 45$

Add $80$ and $45$ .

$25 a = 125$

Divide each term in $25 a = 125$ by $25$ and simplify.

Divide each term in $25 a = 125$ by $25$ .

$\frac{25 a}{25} = \frac{125}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 a}{25} = \frac{125}{25}$

Divide $a$ by $1$ .

$a = \frac{125}{25}$

Simplify the right side.

Divide $125$ by $25$ .

$a = 5$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, - 45)$ and $a = 5$ is $y = (5) {(x - (0))}^{2} - 45$ .

$y = (5) {(x - (0))}^{2} - 45$

Step 4

Solve $y = (5) {(x - (0))}^{2} - 45$ for $y$ .

Remove parentheses.

$y = (5) {(x - (0))}^{2} - 45$

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

$y = 5 {(x - (0))}^{2} - 45$

Remove parentheses.

$y = (5) {(x - (0))}^{2} - 45$

Subtract $0$ from $x$ .

$y = 5 x^{2} - 45$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 5 x^{2} - 45$

Vertex Form: $y = (5) {(x - (0))}^{2} - 45$

Step 6

Simplify the standard form.

Standard Form: $y = 5 x^{2} - 45$

Vertex Form: $y = 5 x^{2} - 45$

Step 7

Find the Parabola Through (-5,80) with Vertex (0,-45) (-5,80) , (0,-45)

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