Graph h(x)=-3/2(x+8)(x+4)

$h (x) = - \frac{3}{2} (x + 8) (x + 4)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Remove parentheses.

$y = - \frac{3}{2} \cdot ((x + 8) (x + 4))$

Remove parentheses.

$y = - \frac{3}{2} \cdot ((x + 8) (x + 4))$

Simplify $- \frac{3}{2} \cdot ((x + 8) (x + 4))$ .

Expand $(x + 8) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$y = - \frac{3}{2} \cdot (x (x + 4) + 8 (x + 4))$

Apply the distributive property.

$y = - \frac{3}{2} \cdot (x \cdot x + x \cdot 4 + 8 (x + 4))$

Apply the distributive property.

$y = - \frac{3}{2} \cdot (x \cdot x + x \cdot 4 + 8 x + 8 \cdot 4)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - \frac{3}{2} \cdot (x^{2} + x \cdot 4 + 8 x + 8 \cdot 4)$

Move $4$ to the left of $x$ .

$y = - \frac{3}{2} \cdot (x^{2} + 4 \cdot x + 8 x + 8 \cdot 4)$

Multiply $8$ by $4$ .

$y = - \frac{3}{2} \cdot (x^{2} + 4 x + 8 x + 32)$

Add $4 x$ and $8 x$ .

$y = - \frac{3}{2} \cdot (x^{2} + 12 x + 32)$

Apply the distributive property.

$y = - \frac{3}{2} x^{2} - \frac{3}{2} (12 x) - \frac{3}{2} \cdot 32$

Simplify.

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

$y = - \frac{x^{2} \cdot 3}{2} - \frac{3}{2} (12 x) - \frac{3}{2} \cdot 32$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = - \frac{x^{2} \cdot 3}{2} + \frac{- 3}{2} (12 x) - \frac{3}{2} \cdot 32$

Factor $2$ out of $12 x$ .

$y = - \frac{x^{2} \cdot 3}{2} + \frac{- 3}{2} (2 (6 x)) - \frac{3}{2} \cdot 32$

Cancel the common factor.

$y = - \frac{x^{2} \cdot 3}{2} + \frac{- 3}{2} (2 (6 x)) - \frac{3}{2} \cdot 32$

Rewrite the expression.

$y = - \frac{x^{2} \cdot 3}{2} - 3 (6 x) - \frac{3}{2} \cdot 32$

Multiply $6$ by $- 3$ .

$y = - \frac{x^{2} \cdot 3}{2} - 18 x - \frac{3}{2} \cdot 32$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = - \frac{x^{2} \cdot 3}{2} - 18 x + \frac{- 3}{2} \cdot 32$

Factor $2$ out of $32$ .

$y = - \frac{x^{2} \cdot 3}{2} - 18 x + \frac{- 3}{2} \cdot (2 (16))$

Cancel the common factor.

$y = - \frac{x^{2} \cdot 3}{2} - 18 x + \frac{- 3}{2} \cdot (2 \cdot 16)$

Rewrite the expression.

$y = - \frac{x^{2} \cdot 3}{2} - 18 x - 3 \cdot 16$

Multiply $- 3$ by $16$ .

$y = - \frac{x^{2} \cdot 3}{2} - 18 x - 48$

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 x^{2}}{2} - 18 x - 48$

Complete the square for $- \frac{3 x^{2}}{2} - 18 x - 48$ .

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

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

$b = - 18$

$c = - 48$

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

$d = \frac{- 18}{2 (- \frac{3}{2})}$

Simplify the right side.

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 3 \cdot - 2$

Multiply $- 3$ by $- 2$ .

$d = 6$

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

$e = - 48 - \frac{{(- 18)}^{2}}{4 (- \frac{3}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = - 48 - \frac{324}{4 (- \frac{3}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 48 - \frac{324}{- 4 (\frac{3}{2})}$

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

$e = - 48 - \frac{324}{\frac{- 4 \cdot 3}{2}}$

Multiply $- 4$ by $3$ .

$e = - 48 - \frac{324}{\frac{- 12}{2}}$

Divide $- 12$ by $2$ .

$e = - 48 - \frac{324}{- 6}$

Divide $324$ by $- 6$ .

$e = - 48 - - 54$

Multiply $- 1$ by $- 54$ .

$e = - 48 + 54$

Add $- 48$ and $54$ .

$e = 6$

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

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

Set $y$ equal to the new right side.

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

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

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

$h = - 6$

$k = 6$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 6, 6)$

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.

$\frac{1}{4 \cdot (- \frac{3}{2})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{3}{2})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{3}{2})}$

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

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

Simplify the expression.

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{2}}$

Divide $12$ by $2$ .

$- \frac{1}{6}$

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.

$(- 6, \frac{35}{6})$

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

$x = - 6$

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.

$y = \frac{37}{6}$

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

Direction: Opens Down

Vertex: $(- 6, 6)$

Focus: $(- 6, \frac{35}{6})$

Axis of Symmetry: $x = - 6$

Directrix: $y = \frac{37}{6}$

Direction: Opens Down

Vertex: $(- 6, 6)$

Focus: $(- 6, \frac{35}{6})$

Axis of Symmetry: $x = - 6$

Directrix: $y = \frac{37}{6}$

Step 2

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 $- 8$ in the expression.

$f (- 8) = - \frac{3 {(- 8)}^{2}}{2} - 18 \cdot - 8 - 48$

Simplify the result.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$f (- 8) = - \frac{3 \cdot 64}{2} - 18 \cdot - 8 - 48$

Multiply $3$ by $64$ .

$f (- 8) = - \frac{192}{2} - 18 \cdot - 8 - 48$

Divide $192$ by $2$ .

$f (- 8) = - 1 \cdot 96 - 18 \cdot - 8 - 48$

Multiply $- 1$ by $96$ .

$f (- 8) = - 96 - 18 \cdot - 8 - 48$

Multiply $- 18$ by $- 8$ .

$f (- 8) = - 96 + 144 - 48$

Simplify by adding and subtracting.

Add $- 96$ and $144$ .

$f (- 8) = 48 - 48$

Subtract $48$ from $48$ .

$f (- 8) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 8$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = - \frac{3 {(- 7)}^{2}}{2} - 18 \cdot - 7 - 48$

Simplify the result.

Simplify each term.

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

$f (- 7) = - \frac{3 \cdot 49}{2} - 18 \cdot - 7 - 48$

Multiply $3$ by $49$ .

$f (- 7) = - \frac{147}{2} - 18 \cdot - 7 - 48$

Multiply $- 18$ by $- 7$ .

$f (- 7) = - \frac{147}{2} + 126 - 48$

Find the common denominator.

Write $126$ as a fraction with denominator $1$ .

$f (- 7) = - \frac{147}{2} + \frac{126}{1} - 48$

Multiply $\frac{126}{1}$ by $\frac{2}{2}$ .

$f (- 7) = - \frac{147}{2} + \frac{126}{1} \cdot \frac{2}{2} - 48$

Multiply $\frac{126}{1}$ by $\frac{2}{2}$ .

$f (- 7) = - \frac{147}{2} + \frac{126 \cdot 2}{2} - 48$

Write $- 48$ as a fraction with denominator $1$ .

$f (- 7) = - \frac{147}{2} + \frac{126 \cdot 2}{2} + \frac{- 48}{1}$

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f (- 7) = - \frac{147}{2} + \frac{126 \cdot 2}{2} + \frac{- 48}{1} \cdot \frac{2}{2}$

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f (- 7) = - \frac{147}{2} + \frac{126 \cdot 2}{2} + \frac{- 48 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 7) = \frac{- 147 + 126 \cdot 2 - 48 \cdot 2}{2}$

Simplify each term.

Multiply $126$ by $2$ .

$f (- 7) = \frac{- 147 + 252 - 48 \cdot 2}{2}$

Multiply $- 48$ by $2$ .

$f (- 7) = \frac{- 147 + 252 - 96}{2}$

Simplify by adding and subtracting.

Add $- 147$ and $252$ .

$f (- 7) = \frac{105 - 96}{2}$

Subtract $96$ from $105$ .

$f (- 7) = \frac{9}{2}$

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = - 7$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - \frac{3 {(- 4)}^{2}}{2} - 18 \cdot - 4 - 48$

Simplify the result.

Simplify each term.

Raise $- 4$ to the power of $2$ .

$f (- 4) = - \frac{3 \cdot 16}{2} - 18 \cdot - 4 - 48$

Multiply $3$ by $16$ .

$f (- 4) = - \frac{48}{2} - 18 \cdot - 4 - 48$

Divide $48$ by $2$ .

$f (- 4) = - 1 \cdot 24 - 18 \cdot - 4 - 48$

Multiply $- 1$ by $24$ .

$f (- 4) = - 24 - 18 \cdot - 4 - 48$

Multiply $- 18$ by $- 4$ .

$f (- 4) = - 24 + 72 - 48$

Simplify by adding and subtracting.

Add $- 24$ and $72$ .

$f (- 4) = 48 - 48$

Subtract $48$ from $48$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - \frac{3 {(- 5)}^{2}}{2} - 18 \cdot - 5 - 48$

Simplify the result.

Simplify each term.

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

$f (- 5) = - \frac{3 \cdot 25}{2} - 18 \cdot - 5 - 48$

Multiply $3$ by $25$ .

$f (- 5) = - \frac{75}{2} - 18 \cdot - 5 - 48$

Multiply $- 18$ by $- 5$ .

$f (- 5) = - \frac{75}{2} + 90 - 48$

Find the common denominator.

Write $90$ as a fraction with denominator $1$ .

$f (- 5) = - \frac{75}{2} + \frac{90}{1} - 48$

Multiply $\frac{90}{1}$ by $\frac{2}{2}$ .

$f (- 5) = - \frac{75}{2} + \frac{90}{1} \cdot \frac{2}{2} - 48$

Multiply $\frac{90}{1}$ by $\frac{2}{2}$ .

$f (- 5) = - \frac{75}{2} + \frac{90 \cdot 2}{2} - 48$

Write $- 48$ as a fraction with denominator $1$ .

$f (- 5) = - \frac{75}{2} + \frac{90 \cdot 2}{2} + \frac{- 48}{1}$

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f (- 5) = - \frac{75}{2} + \frac{90 \cdot 2}{2} + \frac{- 48}{1} \cdot \frac{2}{2}$

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f (- 5) = - \frac{75}{2} + \frac{90 \cdot 2}{2} + \frac{- 48 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 5) = \frac{- 75 + 90 \cdot 2 - 48 \cdot 2}{2}$

Simplify each term.

Multiply $90$ by $2$ .

$f (- 5) = \frac{- 75 + 180 - 48 \cdot 2}{2}$

Multiply $- 48$ by $2$ .

$f (- 5) = \frac{- 75 + 180 - 96}{2}$

Simplify by adding and subtracting.

Add $- 75$ and $180$ .

$f (- 5) = \frac{105 - 96}{2}$

Subtract $96$ from $105$ .

$f (- 5) = \frac{9}{2}$

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

The $y$ value at $x = - 5$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 8 & 0 \\ - 7 & \frac{9}{2} \\ - 6 & 6 \\ - 5 & \frac{9}{2} \\ - 4 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 6, 6)$

Focus: $(- 6, \frac{35}{6})$

Axis of Symmetry: $x = - 6$

Directrix: $y = \frac{37}{6}$

$\begin{array}{c} x & y \\ - 8 & 0 \\ - 7 & \frac{9}{2} \\ - 6 & 6 \\ - 5 & \frac{9}{2} \\ - 4 & 0 \end{array}$

Step 4

Graph h(x)=-3/2(x+8)(x+4)

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