Find the Vertex y=4x^2+5x-1

$y = 4 x^{2} + 5 x - 1$

Step 1

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} + 5 x - 1$ .

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

$a = 4$

$b = 5$

$c = - 1$

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

$d = \frac{5}{2 \cdot 4}$

Multiply $2$ by $4$ .

$d = \frac{5}{8}$

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

$e = - 1 - \frac{5^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $5$ to the power of $2$ .

$e = - 1 - \frac{25}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = - 1 - \frac{25}{16}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$e = - 1 \cdot \frac{16}{16} - \frac{25}{16}$

Combine $- 1$ and $\frac{16}{16}$ .

$e = \frac{- 1 \cdot 16}{16} - \frac{25}{16}$

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 16 - 25}{16}$

Simplify the numerator.

Multiply $- 1$ by $16$ .

$e = \frac{- 16 - 25}{16}$

Subtract $25$ from $- 16$ .

$e = \frac{- 41}{16}$

Move the negative in front of the fraction.

$e = - \frac{41}{16}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$ .

$4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{5}{8})}^{2} - \frac{41}{16}$

Step 2

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

$a = 4$

$h = - \frac{5}{8}$

$k = - \frac{41}{16}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{8}, - \frac{41}{16})$

Step 4

Find the Vertex y=4x^2+5x-1

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