Find the Vertex y=-(x+2)(x-4)

$y = - (x + 2) (x - 4)$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- (x + 2) (x - 4)$ .

Simplify the expression.

Apply the distributive property.

$(- x - 1 \cdot 2) (x - 4)$

Multiply $- 1$ by $2$ .

$(- x - 2) (x - 4)$

Expand $(- x - 2) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$- x (x - 4) - 2 (x - 4)$

Apply the distributive property.

$- x \cdot x - x \cdot - 4 - 2 (x - 4)$

Apply the distributive property.

$- x \cdot x - x \cdot - 4 - 2 x - 2 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$- (x \cdot x) - x \cdot - 4 - 2 x - 2 \cdot - 4$

Multiply $x$ by $x$ .

$- x^{2} - x \cdot - 4 - 2 x - 2 \cdot - 4$

Multiply $- 4$ by $- 1$ .

$- x^{2} + 4 x - 2 x - 2 \cdot - 4$

Multiply $- 2$ by $- 4$ .

$- x^{2} + 4 x - 2 x + 8$

Subtract $2 x$ from $4 x$ .

$- x^{2} + 2 x + 8$

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

$a = - 1$

$b = 2$

$c = 8$

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 $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot - 1}$

Rewrite the expression.

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

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 8 - \frac{2^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = 8 - \frac{4}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 8 - \frac{4}{- 4}$

Divide $4$ by $- 4$ .

$e = 8 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 8 + 1$

Add $8$ and $1$ .

$e = 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 1)}^{2} + 9$ .

$- {(x - 1)}^{2} + 9$

Set $y$ equal to the new right side.

$y = - {(x - 1)}^{2} + 9$

Step 2

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

$a = - 1$

$h = 1$

$k = 9$

Step 3

Find the vertex $(h, k)$ .

$(1, 9)$

Step 4

Find the Vertex y=-(x+2)(x-4)

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