Graph y=-3/2x^2-6x

$y = - \frac{3}{2} x^{2} - 6 x$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify each term.

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

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

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

$y = - \frac{3 x^{2}}{2} - 6 x$

Complete the square for $- \frac{3 x^{2}}{2} - 6 x$ .

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

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

$b = - 6$

$c = 0$

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

$d = \frac{- 6}{2 (- \frac{3}{2})}$

Simplify the right side.

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - - 2$

Multiply $- 1$ by $- 2$ .

$d = 2$

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

$e = 0 - \frac{{(- 6)}^{2}}{4 (- \frac{3}{2})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{36}{4 (- \frac{3}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{36}{- 4 (\frac{3}{2})}$

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

$e = 0 - \frac{36}{\frac{- 4 \cdot 3}{2}}$

Multiply $- 4$ by $3$ .

$e = 0 - \frac{36}{\frac{- 12}{2}}$

Divide $- 12$ by $2$ .

$e = 0 - \frac{36}{- 6}$

Divide $36$ by $- 6$ .

$e = 0 - - 6$

Multiply $- 1$ by $- 6$ .

$e = 0 + 6$

Add $0$ and $6$ .

$e = 6$

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

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

Set $y$ equal to the new right side.

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

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

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

$h = - 2$

$k = 6$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 2, 6)$

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.

$(- 2, \frac{35}{6})$

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

$x = - 2$

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: $(- 2, 6)$

Focus: $(- 2, \frac{35}{6})$

Axis of Symmetry: $x = - 2$

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

Direction: Opens Down

Vertex: $(- 2, 6)$

Focus: $(- 2, \frac{35}{6})$

Axis of Symmetry: $x = - 2$

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 $- 4$ in the expression.

$f (- 4) = - \frac{3 {(- 4)}^{2}}{2} - 6 \cdot - 4$

Simplify the result.

Simplify each term.

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

$f (- 4) = - \frac{3 \cdot 16}{2} - 6 \cdot - 4$

Multiply $3$ by $16$ .

$f (- 4) = - \frac{48}{2} - 6 \cdot - 4$

Divide $48$ by $2$ .

$f (- 4) = - 1 \cdot 24 - 6 \cdot - 4$

Multiply $- 1$ by $24$ .

$f (- 4) = - 24 - 6 \cdot - 4$

Multiply $- 6$ by $- 4$ .

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

Add $- 24$ and $24$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

$f (- 3) = - \frac{3 {(- 3)}^{2}}{2} - 6 \cdot - 3$

Simplify the result.

Simplify each term.

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

$f (- 3) = - \frac{3 \cdot 9}{2} - 6 \cdot - 3$

Multiply $3$ by $9$ .

$f (- 3) = - \frac{27}{2} - 6 \cdot - 3$

Multiply $- 6$ by $- 3$ .

$f (- 3) = - \frac{27}{2} + 18$

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + 18 \cdot \frac{2}{2}$

Combine $18$ and $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{2} + \frac{18 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 27 + 18 \cdot 2}{2}$

Simplify the numerator.

Multiply $18$ by $2$ .

$f (- 3) = \frac{- 27 + 36}{2}$

Add $- 27$ and $36$ .

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

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

$\frac{9}{2}$

The $y$ value at $x = - 3$ is $\frac{9}{2}$ .

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

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{3 {(0)}^{2}}{2} - 6 \cdot 0$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = - \frac{3 \cdot 0}{2} - 6 \cdot 0$

Multiply $3$ by $0$ .

$f (0) = - \frac{0}{2} - 6 \cdot 0$

Divide $0$ by $2$ .

$f (0) = - 0 - 6 \cdot 0$

Multiply $- 1$ by $0$ .

$f (0) = 0 - 6 \cdot 0$

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

$f (- 1) = - \frac{3 {(- 1)}^{2}}{2} - 6 \cdot - 1$

Simplify the result.

Simplify each term.

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

$f (- 1) = - \frac{3 \cdot 1}{2} - 6 \cdot - 1$

Multiply $3$ by $1$ .

$f (- 1) = - \frac{3}{2} - 6 \cdot - 1$

Multiply $- 6$ by $- 1$ .

$f (- 1) = - \frac{3}{2} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 1) = - \frac{3}{2} + 6 \cdot \frac{2}{2}$

Combine $6$ and $\frac{2}{2}$ .

$f (- 1) = - \frac{3}{2} + \frac{6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 3 + 6 \cdot 2}{2}$

Simplify the numerator.

Multiply $6$ by $2$ .

$f (- 1) = \frac{- 3 + 12}{2}$

Add $- 3$ and $12$ .

$f (- 1) = \frac{9}{2}$

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

$\frac{9}{2}$

The $y$ value at $x = - 1$ is $\frac{9}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & \frac{9}{2} \\ - 2 & 6 \\ - 1 & \frac{9}{2} \\ 0 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 2, 6)$

Focus: $(- 2, \frac{35}{6})$

Axis of Symmetry: $x = - 2$

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

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & \frac{9}{2} \\ - 2 & 6 \\ - 1 & \frac{9}{2} \\ 0 & 0 \end{array}$

Step 4

Graph y=-3/2x^2-6x

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