Find the Vertex f(x)=-3x^2-7x+2

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Algebra

$f (x) = - 3 x^{2} - 7 x + 2$

Rewrite the equation in term of $x$ and $y$ .

$f (x) = - 3 x^{2} - 7 x + 2$

Rewrite the equation in vertex form.

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Complete the square for $- 3 x^{2} - 7 x + 2$ .

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

$a = - 3, b = - 7, c = 2$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = - \frac{7}{2 (- 3)}$

Simplify the right side.

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Multiply $2$ by $- 3$ .

$d = - \frac{7}{- 6}$

Move the negative in front of the fraction.

$d = \frac{7}{6}$

Multiply $- - \frac{7}{6}$ .

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Multiply $- 1$ by $- 1$ .

$d = 1 (\frac{7}{6})$

Multiply $\frac{7}{6}$ by $1$ .

$d = \frac{7}{6}$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Raise $- 7$ to the power of $2$ .

$e = 2 - \frac{49}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = 2 - \frac{49}{- 12}$

Move the negative in front of the fraction.

$e = 2 + \frac{49}{12}$

Multiply $- - \frac{49}{12}$ .

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Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{49}{12})$

Multiply $\frac{49}{12}$ by $1$ .

$e = 2 + \frac{49}{12}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = 2 \cdot \frac{12}{12} + \frac{49}{12}$

Combine $2$ and $\frac{12}{12}$ .

$e = \frac{2 \cdot 12}{12} + \frac{49}{12}$

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 12 + 49}{12}$

Simplify the numerator.

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Multiply $2$ by $12$ .

$e = \frac{24 + 49}{12}$

Add $24$ and $49$ .

$e = \frac{73}{12}$

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

$- 3 {(x + \frac{7}{6})}^{2} + \frac{73}{12}$

Set $y$ equal to the new right side.

$y = - 3 {(x + \frac{7}{6})}^{2} + \frac{73}{12}$

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

$a = - 3$

$h = - \frac{7}{6}$

$k = \frac{73}{12}$

Find the vertex $(h, k)$ .

$(- \frac{7}{6}, \frac{73}{12})$

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