Find the Vertex y=-5(x+7)^2-1.3

$y = - 5 {(x + 7)}^{2} - 1.3$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 5 {(x + 7)}^{2} - 1.3$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$- 5 ((x + 7) (x + 7)) - 1.3$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$- 5 (x (x + 7) + 7 (x + 7)) - 1.3$

Apply the distributive property.

$- 5 (x \cdot x + x \cdot 7 + 7 (x + 7)) - 1.3$

Apply the distributive property.

$- 5 (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) - 1.3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 5 (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) - 1.3$

Move $7$ to the left of $x$ .

$- 5 (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) - 1.3$

Multiply $7$ by $7$ .

$- 5 (x^{2} + 7 x + 7 x + 49) - 1.3$

Add $7 x$ and $7 x$ .

$- 5 (x^{2} + 14 x + 49) - 1.3$

Apply the distributive property.

$- 5 x^{2} - 5 (14 x) - 5 \cdot 49 - 1.3$

Simplify.

Multiply $14$ by $- 5$ .

$- 5 x^{2} - 70 x - 5 \cdot 49 - 1.3$

Multiply $- 5$ by $49$ .

$- 5 x^{2} - 70 x - 245 - 1.3$

Subtract $1.3$ from $- 245$ .

$- 5 x^{2} - 70 x - 246.3$

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 5$

$b = - 70$

$c = - 246.3$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 70}{2 \cdot - 5}$

Simplify the right side.

Cancel the common factor of $- 70$ and $2$ .

Factor $2$ out of $- 70$ .

$d = \frac{2 \cdot - 35}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$d = \frac{2 \cdot - 35}{2 (- 5)}$

Cancel the common factor.

$d = \frac{2 \cdot - 35}{2 \cdot - 5}$

Rewrite the expression.

$d = \frac{- 35}{- 5}$

Cancel the common factor of $- 35$ and $- 5$ .

Factor $- 5$ out of $- 35$ .

$d = \frac{- 5 (7)}{- 5}$

Cancel the common factors.

Factor $- 5$ out of $- 5$ .

$d = \frac{- 5 \cdot 7}{- 5 \cdot 1}$

Cancel the common factor.

$d = \frac{- 5 \cdot 7}{- 5 \cdot 1}$

Rewrite the expression.

$d = \frac{7}{1}$

Divide $7$ by $1$ .

$d = 7$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 246.3 - \frac{{(- 70)}^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $- 70$ to the power of $2$ .

$e = - 246.3 - \frac{4900}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = - 246.3 - \frac{4900}{- 20}$

Divide $4900$ by $- 20$ .

$e = - 246.3 - - 245$

Multiply $- 1$ by $- 245$ .

$e = - 246.3 + 245$

Add $- 246.3$ and $245$ .

$e = - 1.3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x + 7)}^{2} - 1.3$ .

$- 5 {(x + 7)}^{2} - 1.3$

Set $y$ equal to the new right side.

$y = - 5 {(x + 7)}^{2} - 1.3$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 5$

$h = - 7$

$k = - 1.3$

Step 3

Find the vertex $(h, k)$ .

$(- 7, - 1.3)$

Step 4

Find the Vertex y=-5(x+7)^2-1.3

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