Find the Vertex Form y=9/4x^2+3x-1

$y = \frac{9}{4} x^{2} + 3 x - 1$

Step 1

Combine $\frac{9}{4}$ and $x^{2}$ .

$y = \frac{9 x^{2}}{4} + 3 x - 1$

Step 2

Complete the square for $\frac{9 x^{2}}{4} + 3 x - 1$ .

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

$a = \frac{9}{4}$

$b = 3$

$c = - 1$

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

$d = \frac{3}{2 (\frac{9}{4})}$

Simplify the right side.

Combine $2$ and $\frac{9}{4}$ .

$d = \frac{3}{\frac{2 \cdot 9}{4}}$

Multiply $2$ by $9$ .

$d = \frac{3}{\frac{18}{4}}$

Cancel the common factor of $18$ and $4$ .

Factor $2$ out of $18$ .

$d = \frac{3}{\frac{2 (9)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$d = \frac{3}{\frac{2 \cdot 9}{2 \cdot 2}}$

Cancel the common factor.

$d = \frac{3}{\frac{2 \cdot 9}{2 \cdot 2}}$

Rewrite the expression.

$d = \frac{3}{\frac{9}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 3 (\frac{2}{9})$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$d = 3 \frac{2}{3 (3)}$

Cancel the common factor.

$d = 3 \frac{2}{3 \cdot 3}$

Rewrite the expression.

$d = \frac{2}{3}$

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

$e = - 1 - \frac{3^{2}}{4 (\frac{9}{4})}$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = - 1 - \frac{9}{4 (\frac{9}{4})}$

Combine $4$ and $\frac{9}{4}$ .

$e = - 1 - \frac{9}{\frac{4 \cdot 9}{4}}$

Multiply $4$ by $9$ .

$e = - 1 - \frac{9}{\frac{36}{4}}$

Divide $36$ by $4$ .

$e = - 1 - \frac{9}{9}$

Cancel the common factor of $9$ .

Cancel the common factor.

$e = - 1 - \frac{9}{9}$

Rewrite the expression.

$e = - 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = - 1 - 1$

Subtract $1$ from $- 1$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{9}{4} {(x + \frac{2}{3})}^{2} - 2$ .

$\frac{9}{4} {(x + \frac{2}{3})}^{2} - 2$

Step 3

Set $y$ equal to the new right side.

$y = \frac{9}{4} \cdot {(x + \frac{2}{3})}^{2} - 2$

Find the Vertex Form y=9/4x^2+3x-1

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