Find the Vertex -4x^2-4x+1

$- 4 x^{2} - 4 x + 1$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 4 x^{2} - 4 x + 1$ .

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

$a = - 4$

$b = - 4$

$c = 1$

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

$d = \frac{- 4}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot - 2}{2 (- 4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{- 2}{- 4}$

Cancel the common factor of $- 2$ and $- 4$ .

Factor $- 2$ out of $- 2$ .

$d = \frac{- 2 (1)}{- 4}$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $- 4$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Cancel the common factors.

Factor $- 4$ out of $4 \cdot - 4$ .

$e = 1 - \frac{- 4 \cdot - 4}{- 4 (- - 4)}$

Cancel the common factor.

$e = 1 - \frac{- 4 \cdot - 4}{- 4 (- - 4)}$

Rewrite the expression.

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

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$e = 1 - (- 1 \cdot 1)$

Multiply $- (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$e = 1 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 1 + 1$

Add $1$ and $1$ .

$e = 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + \frac{1}{2})}^{2} + 2$ .

$- 4 {(x + \frac{1}{2})}^{2} + 2$

Set $y$ equal to the new right side.

$y = - 4 {(x + \frac{1}{2})}^{2} + 2$

Step 2

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

$a = - 4$

$h = - \frac{1}{2}$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{1}{2}, 2)$

Step 4

Find the Vertex -4x^2-4x+1

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