Find the Parabola Through (4,-5) with Vertex (-1,5) (-1,5) , (4,-5)

$(- 1, 5)$ , $(4, - 5)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 1, 5)$ as the vertex $(h, k)$ and $(4, - 5)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 5 = a {(4 - (- 1))}^{2} + 5$

Step 2

Using $- 5 = a {(4 - (- 1))}^{2} + 5$ to solve for $a$ , $a = - \frac{2}{5}$ .

Rewrite the equation as $a {(4 - (- 1))}^{2} + 5 = - 5$ .

$a {(4 - (- 1))}^{2} + 5 = - 5$

Simplify each term.

Multiply $- 1$ by $- 1$ .

$a {(4 + 1)}^{2} + 5 = - 5$

Add $4$ and $1$ .

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

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

$a \cdot 25 + 5 = - 5$

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

$25 a + 5 = - 5$

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

Subtract $5$ from both sides of the equation.

$25 a = - 5 - 5$

Subtract $5$ from $- 5$ .

$25 a = - 10$

Divide each term in $25 a = - 10$ by $25$ and simplify.

Divide each term in $25 a = - 10$ by $25$ .

$\frac{25 a}{25} = \frac{- 10}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 a}{25} = \frac{- 10}{25}$

Divide $a$ by $1$ .

$a = \frac{- 10}{25}$

Simplify the right side.

Cancel the common factor of $- 10$ and $25$ .

Factor $5$ out of $- 10$ .

$a = \frac{5 (- 2)}{25}$

Cancel the common factors.

Factor $5$ out of $25$ .

$a = \frac{5 \cdot - 2}{5 \cdot 5}$

Cancel the common factor.

$a = \frac{5 \cdot - 2}{5 \cdot 5}$

Rewrite the expression.

$a = \frac{- 2}{5}$

Move the negative in front of the fraction.

$a = - \frac{2}{5}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 1, 5)$ and $a = - \frac{2}{5}$ is $y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$ .

$y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$

Step 4

Solve $y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$ for $y$ .

Remove parentheses.

$y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$

Multiply $- \frac{2}{5}$ by ${(x - (- 1))}^{2}$ .

$y = - \frac{2}{5} \cdot {(x - (- 1))}^{2} + 5$

Remove parentheses.

$y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$

Simplify $(- \frac{2}{5}) {(x - (- 1))}^{2} + 5$ .

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = - \frac{2}{5} {(x + 1)}^{2} + 5$

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$y = - \frac{2}{5} ((x + 1) (x + 1)) + 5$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = - \frac{2}{5} (x (x + 1) + 1 (x + 1)) + 5$

Apply the distributive property.

$y = - \frac{2}{5} (x \cdot x + x \cdot 1 + 1 (x + 1)) + 5$

Apply the distributive property.

$y = - \frac{2}{5} (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - \frac{2}{5} (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 5$

Multiply $x$ by $1$ .

$y = - \frac{2}{5} (x^{2} + x + 1 x + 1 \cdot 1) + 5$

Multiply $x$ by $1$ .

$y = - \frac{2}{5} (x^{2} + x + x + 1 \cdot 1) + 5$

Multiply $1$ by $1$ .

$y = - \frac{2}{5} (x^{2} + x + x + 1) + 5$

Add $x$ and $x$ .

$y = - \frac{2}{5} (x^{2} + 2 x + 1) + 5$

Apply the distributive property.

$y = - \frac{2}{5} x^{2} - \frac{2}{5} (2 x) - \frac{2}{5} \cdot 1 + 5$

Simplify.

Combine $x^{2}$ and $\frac{2}{5}$ .

$y = - \frac{x^{2} \cdot 2}{5} - \frac{2}{5} (2 x) - \frac{2}{5} \cdot 1 + 5$

Multiply $- \frac{2}{5} (2 x)$ .

Multiply $2$ by $- 1$ .

$y = - \frac{x^{2} \cdot 2}{5} - 2 (\frac{2}{5}) x - \frac{2}{5} \cdot 1 + 5$

Combine $- 2$ and $\frac{2}{5}$ .

$y = - \frac{x^{2} \cdot 2}{5} + \frac{- 2 \cdot 2}{5} x - \frac{2}{5} \cdot 1 + 5$

Multiply $- 2$ by $2$ .

$y = - \frac{x^{2} \cdot 2}{5} + \frac{- 4}{5} x - \frac{2}{5} \cdot 1 + 5$

Combine $\frac{- 4}{5}$ and $x$ .

$y = - \frac{x^{2} \cdot 2}{5} + \frac{- 4 x}{5} - \frac{2}{5} \cdot 1 + 5$

Multiply $- 1$ by $1$ .

$y = - \frac{x^{2} \cdot 2}{5} + \frac{- 4 x}{5} - \frac{2}{5} + 5$

Simplify each term.

Move $2$ to the left of $x^{2}$ .

$y = - \frac{2 \cdot x^{2}}{5} + \frac{- 4 x}{5} - \frac{2}{5} + 5$

Move the negative in front of the fraction.

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} - \frac{2}{5} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} - \frac{2}{5} + 5 \cdot \frac{5}{5}$

Combine $5$ and $\frac{5}{5}$ .

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} - \frac{2}{5} + \frac{5 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} + \frac{- 2 + 5 \cdot 5}{5}$

Simplify the numerator.

Multiply $5$ by $5$ .

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} + \frac{- 2 + 25}{5}$

Add $- 2$ and $25$ .

$y = - \frac{2 x^{2}}{5} - \frac{4 x}{5} + \frac{23}{5}$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - \frac{2}{5} x^{2} - \frac{4}{5} x + \frac{23}{5}$

Vertex Form: $y = (- \frac{2}{5}) {(x - (- 1))}^{2} + 5$

Step 6

Simplify the standard form.

Standard Form: $y = - \frac{2}{5} x^{2} - \frac{4}{5} x + \frac{23}{5}$

Vertex Form: $y = - \frac{2}{5} {(x + 1)}^{2} + 5$

Step 7

Find the Parabola Through (4,-5) with Vertex (-1,5) (-1,5) , (4,-5)

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