Graph 2/3(x+7)(x+1)

$\frac{2}{3} (x + 7) (x + 1)$

Solve for $y$ .

Remove parentheses.

$y = \frac{2}{3} \cdot ((x + 7) (x + 1))$

Remove parentheses.

$y = \frac{2}{3} \cdot ((x + 7) (x + 1))$

Simplify $\frac{2}{3} \cdot ((x + 7) (x + 1))$ .

Expand $(x + 7) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{2}{3} \cdot (x (x + 1) + 7 (x + 1))$

Apply the distributive property.

$y = \frac{2}{3} \cdot (x \cdot x + x \cdot 1 + 7 (x + 1))$

Apply the distributive property.

$y = \frac{2}{3} \cdot (x \cdot x + x \cdot 1 + 7 x + 7 \cdot 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \frac{2}{3} \cdot (x^{2} + x \cdot 1 + 7 x + 7 \cdot 1)$

Multiply $x$ by $1$ .

$y = \frac{2}{3} \cdot (x^{2} + x + 7 x + 7 \cdot 1)$

Multiply $7$ by $1$ .

$y = \frac{2}{3} \cdot (x^{2} + x + 7 x + 7)$

Add $x$ and $7 x$ .

$y = \frac{2}{3} \cdot (x^{2} + 8 x + 7)$

Apply the distributive property.

$y = \frac{2}{3} x^{2} + \frac{2}{3} (8 x) + \frac{2}{3} \cdot 7$

Simplify.

Combine $\frac{2}{3}$ and $x^{2}$ .

$y = \frac{2 x^{2}}{3} + \frac{2}{3} (8 x) + \frac{2}{3} \cdot 7$

Multiply $\frac{2}{3} (8 x)$ .

Combine $8$ and $\frac{2}{3}$ .

$y = \frac{2 x^{2}}{3} + \frac{8 \cdot 2}{3} x + \frac{2}{3} \cdot 7$

Multiply $8$ by $2$ .

$y = \frac{2 x^{2}}{3} + \frac{16}{3} x + \frac{2}{3} \cdot 7$

Combine $\frac{16}{3}$ and $x$ .

$y = \frac{2 x^{2}}{3} + \frac{16 x}{3} + \frac{2}{3} \cdot 7$

Multiply $\frac{2}{3} \cdot 7$ .

Combine $\frac{2}{3}$ and $7$ .

$y = \frac{2 x^{2}}{3} + \frac{16 x}{3} + \frac{2 \cdot 7}{3}$

Multiply $2$ by $7$ .

$y = \frac{2 x^{2}}{3} + \frac{16 x}{3} + \frac{14}{3}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{2 x^{2}}{3} + \frac{16 x}{3} + \frac{14}{3}$ .

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

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

$b = \frac{16}{3}$

$c = \frac{14}{3}$

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

$d = \frac{\frac{16}{3}}{2 (\frac{2}{3})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{16}{3} \cdot \frac{1}{2 (\frac{2}{3})}$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$d = \frac{2 (8)}{3} \cdot \frac{1}{2 (\frac{2}{3})}$

Cancel the common factor.

$d = \frac{2 \cdot 8}{3} \cdot \frac{1}{2 (\frac{2}{3})}$

Rewrite the expression.

$d = \frac{8}{3} \cdot \frac{1}{\frac{2}{3}}$

Multiply $\frac{8}{3}$ by $\frac{1}{\frac{2}{3}}$ .

$d = \frac{8}{3 (\frac{2}{3})}$

Combine $3$ and $\frac{2}{3}$ .

$d = \frac{8}{\frac{3 \cdot 2}{3}}$

Multiply $3$ by $2$ .

$d = \frac{8}{\frac{6}{3}}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

$d = \frac{8}{\frac{3 \cdot 2}{3}}$

Cancel the common factors.

Factor $3$ out of $3$ .

$d = \frac{8}{\frac{3 \cdot 2}{3 (1)}}$

Cancel the common factor.

$d = \frac{8}{\frac{3 \cdot 2}{3 \cdot 1}}$

Rewrite the expression.

$d = \frac{8}{\frac{2}{1}}$

Divide $2$ by $1$ .

$d = \frac{8}{2}$

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 1}$

Rewrite the expression.

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

Divide $4$ by $1$ .

$d = 4$

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

$e = \frac{14}{3} - \frac{{(\frac{16}{3})}^{2}}{4 (\frac{2}{3})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{16}{3}$ .

$e = \frac{14}{3} - \frac{\frac{16^{2}}{3^{2}}}{4 (\frac{2}{3})}$

Raise $16$ to the power of $2$ .

$e = \frac{14}{3} - \frac{\frac{256}{3^{2}}}{4 (\frac{2}{3})}$

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

$e = \frac{14}{3} - \frac{\frac{256}{9}}{4 (\frac{2}{3})}$

Combine $4$ and $\frac{2}{3}$ .

$e = \frac{14}{3} - \frac{\frac{256}{9}}{\frac{4 \cdot 2}{3}}$

Multiply $4$ by $2$ .

$e = \frac{14}{3} - \frac{\frac{256}{9}}{\frac{8}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{14}{3} - (\frac{256}{9} \cdot \frac{3}{8})$

Cancel the common factor of $8$ .

Factor $8$ out of $256$ .

$e = \frac{14}{3} - (\frac{8 (32)}{9} \cdot \frac{3}{8})$

Cancel the common factor.

$e = \frac{14}{3} - (\frac{8 \cdot 32}{9} \cdot \frac{3}{8})$

Rewrite the expression.

$e = \frac{14}{3} - (\frac{32}{9} \cdot 3)$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = \frac{14}{3} - (\frac{32}{3 (3)} \cdot 3)$

Cancel the common factor.

$e = \frac{14}{3} - (\frac{32}{3 \cdot 3} \cdot 3)$

Rewrite the expression.

$e = \frac{14}{3} - \frac{32}{3}$

Combine the numerators over the common denominator.

$e = \frac{14 - 32}{3}$

Subtract $32$ from $14$ .

$e = \frac{- 18}{3}$

Divide $- 18$ by $3$ .

$e = - 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{2}{3} {(x + 4)}^{2} - 6$ .

$\frac{2}{3} {(x + 4)}^{2} - 6$

Set $y$ equal to the new right side.

$y = \frac{2}{3} \cdot {(x + 4)}^{2} - 6$

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

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

$h = - 4$

$k = - 6$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(- 4, - 6)$

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.

$\frac{1}{4 \cdot \frac{2}{3}}$

Simplify.

Combine $4$ and $\frac{2}{3}$ .

$\frac{1}{\frac{4 \cdot 2}{3}}$

Multiply $4$ by $2$ .

$\frac{1}{\frac{8}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{8})$

Multiply $\frac{3}{8}$ by $1$ .

$\frac{3}{8}$

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.

$(- 4, - \frac{45}{8})$

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

$x = - 4$

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.

$y = - \frac{51}{8}$

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

Direction: Opens Up

Vertex: $(- 4, - 6)$

Focus: $(- 4, - \frac{45}{8})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{51}{8}$

Direction: Opens Up

Vertex: $(- 4, - 6)$

Focus: $(- 4, - \frac{45}{8})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{51}{8}$

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 $- 5$ in the expression.

$f (- 5) = \frac{2 {(- 5)}^{2}}{3} + \frac{16 (- 5)}{3} + \frac{14}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 5) = \frac{2 {(- 5)}^{2} + 16 (- 5) + 14}{3}$

Simplify each term.

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

$f (- 5) = \frac{2 \cdot 25 + 16 (- 5) + 14}{3}$

Multiply $2$ by $25$ .

$f (- 5) = \frac{50 + 16 (- 5) + 14}{3}$

Multiply $16$ by $- 5$ .

$f (- 5) = \frac{50 - 80 + 14}{3}$

Simplify the expression.

Subtract $80$ from $50$ .

$f (- 5) = \frac{- 30 + 14}{3}$

Add $- 30$ and $14$ .

$f (- 5) = \frac{- 16}{3}$

Move the negative in front of the fraction.

$f (- 5) = - \frac{16}{3}$

The final answer is $- \frac{16}{3}$ .

$- \frac{16}{3}$

The $y$ value at $x = - 5$ is $- \frac{16}{3}$ .

$y = - \frac{16}{3}$

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = \frac{2 {(- 6)}^{2}}{3} + \frac{16 (- 6)}{3} + \frac{14}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 6) = \frac{2 {(- 6)}^{2} + 16 (- 6) + 14}{3}$

Simplify each term.

Raise $- 6$ to the power of $2$ .

$f (- 6) = \frac{2 \cdot 36 + 16 (- 6) + 14}{3}$

Multiply $2$ by $36$ .

$f (- 6) = \frac{72 + 16 (- 6) + 14}{3}$

Multiply $16$ by $- 6$ .

$f (- 6) = \frac{72 - 96 + 14}{3}$

Simplify the expression.

Subtract $96$ from $72$ .

$f (- 6) = \frac{- 24 + 14}{3}$

Add $- 24$ and $14$ .

$f (- 6) = \frac{- 10}{3}$

Move the negative in front of the fraction.

$f (- 6) = - \frac{10}{3}$

The final answer is $- \frac{10}{3}$ .

$- \frac{10}{3}$

The $y$ value at $x = - 6$ is $- \frac{10}{3}$ .

$y = - \frac{10}{3}$

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

$f (- 1) = \frac{2 {(- 1)}^{2}}{3} + \frac{16 (- 1)}{3} + \frac{14}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 1) = \frac{2 {(- 1)}^{2} + 16 (- 1) + 14}{3}$

Simplify each term.

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

$f (- 1) = \frac{2 \cdot 1 + 16 (- 1) + 14}{3}$

Multiply $2$ by $1$ .

$f (- 1) = \frac{2 + 16 (- 1) + 14}{3}$

Multiply $16$ by $- 1$ .

$f (- 1) = \frac{2 - 16 + 14}{3}$

Simplify the expression.

Subtract $16$ from $2$ .

$f (- 1) = \frac{- 14 + 14}{3}$

Add $- 14$ and $14$ .

$f (- 1) = \frac{0}{3}$

Divide $0$ by $3$ .

$f (- 1) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 1$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{2 {(- 3)}^{2}}{3} + \frac{16 (- 3)}{3} + \frac{14}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 3) = \frac{2 {(- 3)}^{2} + 16 (- 3) + 14}{3}$

Simplify each term.

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

$f (- 3) = \frac{2 \cdot 9 + 16 (- 3) + 14}{3}$

Multiply $2$ by $9$ .

$f (- 3) = \frac{18 + 16 (- 3) + 14}{3}$

Multiply $16$ by $- 3$ .

$f (- 3) = \frac{18 - 48 + 14}{3}$

Simplify the expression.

Subtract $48$ from $18$ .

$f (- 3) = \frac{- 30 + 14}{3}$

Add $- 30$ and $14$ .

$f (- 3) = \frac{- 16}{3}$

Move the negative in front of the fraction.

$f (- 3) = - \frac{16}{3}$

The final answer is $- \frac{16}{3}$ .

$- \frac{16}{3}$

The $y$ value at $x = - 3$ is $- \frac{16}{3}$ .

$y = - \frac{16}{3}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & - \frac{10}{3} \\ - 5 & - \frac{16}{3} \\ - 4 & - 6 \\ - 3 & - \frac{16}{3} \\ - 1 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 4, - 6)$

Focus: $(- 4, - \frac{45}{8})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{51}{8}$

$\begin{array}{c} x & y \\ - 6 & - \frac{10}{3} \\ - 5 & - \frac{16}{3} \\ - 4 & - 6 \\ - 3 & - \frac{16}{3} \\ - 1 & 0 \end{array}$

Graph 2/3(x+7)(x+1)

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