Graph g(n)=-(3n-1)(2n+1)

$g (n) = - (3 n - 1) (2 n + 1)$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- (3 x - 1) (2 x + 1)$ .

Simplify the expression.

Apply the distributive property.

$(- (3 x) - - 1) (2 x + 1)$

Multiply $3$ by $- 1$ .

$(- 3 x - - 1) (2 x + 1)$

Multiply $- 1$ by $- 1$ .

$(- 3 x + 1) (2 x + 1)$

Expand $(- 3 x + 1) (2 x + 1)$ using the FOIL Method.

Apply the distributive property.

$- 3 x (2 x + 1) + 1 (2 x + 1)$

Apply the distributive property.

$- 3 x (2 x) - 3 x \cdot 1 + 1 (2 x + 1)$

Apply the distributive property.

$- 3 x (2 x) - 3 x \cdot 1 + 1 (2 x) + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 3 \cdot 2 x \cdot x - 3 x \cdot 1 + 1 (2 x) + 1 \cdot 1$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$- 3 \cdot 2 (x \cdot x) - 3 x \cdot 1 + 1 (2 x) + 1 \cdot 1$

Multiply $x$ by $x$ .

$- 3 \cdot 2 x^{2} - 3 x \cdot 1 + 1 (2 x) + 1 \cdot 1$

Multiply $- 3$ by $2$ .

$- 6 x^{2} - 3 x \cdot 1 + 1 (2 x) + 1 \cdot 1$

Multiply $- 3$ by $1$ .

$- 6 x^{2} - 3 x + 1 (2 x) + 1 \cdot 1$

Multiply $2 x$ by $1$ .

$- 6 x^{2} - 3 x + 2 x + 1 \cdot 1$

Multiply $1$ by $1$ .

$- 6 x^{2} - 3 x + 2 x + 1$

Add $- 3 x$ and $2 x$ .

$- 6 x^{2} - x + 1$

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

$a = - 6$

$b = - 1$

$c = 1$

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

$d = \frac{- 1}{2 \cdot - 6}$

Simplify the right side.

Multiply $2$ by $- 6$ .

$d = \frac{- 1}{- 12}$

Dividing two negative values results in a positive value.

$d = \frac{1}{12}$

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

$e = 1 - \frac{{(- 1)}^{2}}{4 \cdot - 6}$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$e = 1 - \frac{1}{4 \cdot - 6}$

Multiply $4$ by $- 6$ .

$e = 1 - \frac{1}{- 24}$

Move the negative in front of the fraction.

$e = 1 - - \frac{1}{24}$

Multiply $- - \frac{1}{24}$ .

Multiply $- 1$ by $- 1$ .

$e = 1 + 1 (\frac{1}{24})$

Multiply $\frac{1}{24}$ by $1$ .

$e = 1 + \frac{1}{24}$

Write $1$ as a fraction with a common denominator.

$e = \frac{24}{24} + \frac{1}{24}$

Combine the numerators over the common denominator.

$e = \frac{24 + 1}{24}$

Add $24$ and $1$ .

$e = \frac{25}{24}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x + \frac{1}{12})}^{2} + \frac{25}{24}$ .

$- 6 {(x + \frac{1}{12})}^{2} + \frac{25}{24}$

Set $y$ equal to the new right side.

$y = - 6 {(x + \frac{1}{12})}^{2} + \frac{25}{24}$

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

$a = - 6$

$h = - \frac{1}{12}$

$k = \frac{25}{24}$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(- \frac{1}{12}, \frac{25}{24})$

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 6}$

Simplify.

Multiply $4$ by $- 6$ .

$\frac{1}{- 24}$

Move the negative in front of the fraction.

$- \frac{1}{24}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- \frac{1}{12}, 1)$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - \frac{1}{12}$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{13}{12}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(- \frac{1}{12}, \frac{25}{24})$

Focus: $(- \frac{1}{12}, 1)$

Axis of Symmetry: $x = - \frac{1}{12}$

Directrix: $y = \frac{13}{12}$

Direction: Opens Down

Vertex: $(- \frac{1}{12}, \frac{25}{24})$

Focus: $(- \frac{1}{12}, 1)$

Axis of Symmetry: $x = - \frac{1}{12}$

Directrix: $y = \frac{13}{12}$

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - (3 (- 1) - 1) (2 (- 1) + 1)$

Simplify the result.

Multiply $3$ by $- 1$ .

$f (- 1) = - (- 3 - 1) (2 (- 1) + 1)$

Subtract $1$ from $- 3$ .

$f (- 1) = 4 (2 (- 1) + 1)$

Multiply $2$ by $- 1$ .

$f (- 1) = 4 (- 2 + 1)$

Add $- 2$ and $1$ .

$f (- 1) = 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$f (- 1) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = - 1$ is $- 4$ .

$y = - 4$

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - (3 (- 2) - 1) (2 (- 2) + 1)$

Simplify the result.

Multiply $3$ by $- 2$ .

$f (- 2) = - (- 6 - 1) (2 (- 2) + 1)$

Subtract $1$ from $- 6$ .

$f (- 2) = 7 (2 (- 2) + 1)$

Multiply $2$ by $- 2$ .

$f (- 2) = 7 (- 4 + 1)$

Add $- 4$ and $1$ .

$f (- 2) = 7 \cdot - 3$

Multiply $7$ by $- 3$ .

$f (- 2) = - 21$

The final answer is $- 21$ .

$- 21$

The $y$ value at $x = - 2$ is $- 21$ .

$y = - 21$

Replace the variable $x$ with $1$ in the expression.

$f (1) = - (3 (1) - 1) (2 (1) + 1)$

Simplify the result.

Multiply $3$ by $1$ .

$f (1) = - (3 - 1) (2 (1) + 1)$

Subtract $1$ from $3$ .

$f (1) = - 1 \cdot (2 (2 (1) + 1))$

Multiply $- 1$ by $2$ .

$f (1) = - 2 (2 (1) + 1)$

Multiply $2$ by $1$ .

$f (1) = - 2 (2 + 1)$

Add $2$ and $1$ .

$f (1) = - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$f (1) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = 1$ is $- 6$ .

$y = - 6$

Replace the variable $x$ with $2$ in the expression.

$f (2) = - (3 (2) - 1) (2 (2) + 1)$

Simplify the result.

Multiply $3$ by $2$ .

$f (2) = - (6 - 1) (2 (2) + 1)$

Subtract $1$ from $6$ .

$f (2) = - 1 \cdot (5 (2 (2) + 1))$

Multiply $- 1$ by $5$ .

$f (2) = - 5 (2 (2) + 1)$

Multiply $2$ by $2$ .

$f (2) = - 5 (4 + 1)$

Add $4$ and $1$ .

$f (2) = - 5 \cdot 5$

Multiply $- 5$ by $5$ .

$f (2) = - 25$

The final answer is $- 25$ .

$- 25$

The $y$ value at $x = 2$ is $- 25$ .

$y = - 25$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - 21 \\ - 1 & - 4 \\ - \frac{1}{12} & \frac{25}{24} \\ 1 & - 6 \\ 2 & - 25 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{1}{12}, \frac{25}{24})$

Focus: $(- \frac{1}{12}, 1)$

Axis of Symmetry: $x = - \frac{1}{12}$

Directrix: $y = \frac{13}{12}$

$\begin{array}{c} x & y \\ - 2 & - 21 \\ - 1 & - 4 \\ - \frac{1}{12} & \frac{25}{24} \\ 1 & - 6 \\ 2 & - 25 \end{array}$

Graph g(n)=-(3n-1)(2n+1)

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