Find the Vertex f(x)=4x^2-8x+1

$f (x) = 4 x^{2} - 8 x + 1$

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} - 8 x + 1$ .

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

$a = 4$

$b = - 8$

$c = 1$

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

$d = \frac{- 8}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $4$ out of $- 4$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ 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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$e = 1 - \frac{64}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 1 - \frac{64}{16}$

Divide $64$ by $16$ .

$e = 1 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 1 - 4$

Subtract $4$ from $1$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 1)}^{2} - 3$ .

$4 {(x - 1)}^{2} - 3$

Set $y$ equal to the new right side.

$y = 4 {(x - 1)}^{2} - 3$

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

$a = 4$

$h = 1$

$k = - 3$

Find the vertex $(h, k)$ .

$(1, - 3)$

Find the Vertex f(x)=4x^2-8x+1

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