Find the Vertex -1/4x^2+2x-3

$- \frac{1}{4} x^{2} + 2 x - 3$

Step 1

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{4}$ .

$y = - \frac{x^{2}}{4} + 2 x - 3$

Complete the square for $- \frac{x^{2}}{4} + 2 x - 3$ .

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

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

$b = 2$

$c = - 3$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 4)$

Multiply $- (1 \cdot 4)$ .

Multiply $4$ by $1$ .

$d = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$d = - 4$

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

$e = - 3 - \frac{2^{2}}{4 (- \frac{1}{4})}$

Simplify the right side.

Simplify each term.

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

$e = - 3 - \frac{4}{4 (- \frac{1}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 3 - \frac{4}{- 4 (\frac{1}{4})}$

Combine $- 4$ and $\frac{1}{4}$ .

$e = - 3 - \frac{4}{\frac{- 4}{4}}$

Divide $- 4$ by $4$ .

$e = - 3 - \frac{4}{- 1}$

Divide $4$ by $- 1$ .

$e = - 3 - - 4$

Multiply $- 1$ by $- 4$ .

$e = - 3 + 4$

Add $- 3$ and $4$ .

$e = 1$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 4$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(4, 1)$

Step 4

Find the Vertex -1/4x^2+2x-3

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