Graph 2x^2+3x+y+1=0

$2 x^{2} + 3 x + y + 1 = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

$3 x + y + 1 = - 2 x^{2}$

Subtract $3 x$ from both sides of the equation.

$y + 1 = - 2 x^{2} - 3 x$

Subtract $1$ from both sides of the equation.

$y = - 2 x^{2} - 3 x - 1$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} - 3 x - 1$ .

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

$a = - 2$

$b = - 3$

$c = - 1$

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

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

Simplify the right side.

Multiply $2$ by $- 2$ .

$d = \frac{- 3}{- 4}$

Dividing two negative values results in a positive value.

$d = \frac{3}{4}$

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

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

Simplify the right side.

Simplify each term.

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

$e = - 1 - \frac{9}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 1 - \frac{9}{- 8}$

Move the negative in front of the fraction.

$e = - 1 - - \frac{9}{8}$

Multiply $- - \frac{9}{8}$ .

Multiply $- 1$ by $- 1$ .

$e = - 1 + 1 (\frac{9}{8})$

Multiply $\frac{9}{8}$ by $1$ .

$e = - 1 + \frac{9}{8}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 1 \cdot \frac{8}{8} + \frac{9}{8}$

Combine $- 1$ and $\frac{8}{8}$ .

$e = \frac{- 1 \cdot 8}{8} + \frac{9}{8}$

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 8 + 9}{8}$

Simplify the numerator.

Multiply $- 1$ by $8$ .

$e = \frac{- 8 + 9}{8}$

Add $- 8$ and $9$ .

$e = \frac{1}{8}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + \frac{3}{4})}^{2} + \frac{1}{8}$ .

$- 2 {(x + \frac{3}{4})}^{2} + \frac{1}{8}$

Set $y$ equal to the new right side.

$y = - 2 {(x + \frac{3}{4})}^{2} + \frac{1}{8}$

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

$a = - 2$

$h = - \frac{3}{4}$

$k = \frac{1}{8}$

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

Opens Down

Find the vertex $(h, k)$ .

$(- \frac{3}{4}, \frac{1}{8})$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- \frac{3}{4}, 0)$

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

$x = - \frac{3}{4}$

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.

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

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

Direction: Opens Down

Vertex: $(- \frac{3}{4}, \frac{1}{8})$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $x = - \frac{3}{4}$

Directrix: $y = \frac{1}{4}$

Direction: Opens Down

Vertex: $(- \frac{3}{4}, \frac{1}{8})$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $x = - \frac{3}{4}$

Directrix: $y = \frac{1}{4}$

Step 3

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 $- 2$ in the expression.

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

Simplify the result.

Simplify each term.

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

Multiply $- 2$ by ${(- 2)}^{2}$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

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

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

$f (- 2) = - 8 - 3 \cdot - 2 - 1$

Multiply $- 3$ by $- 2$ .

$f (- 2) = - 8 + 6 - 1$

Simplify by adding and subtracting.

Add $- 8$ and $6$ .

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

Subtract $1$ from $- 2$ .

$f (- 2) = - 3$

The final answer is $- 3$ .

$- 3$

The $y$ value at $x = - 2$ is $- 3$ .

$y = - 3$

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

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

Simplify the result.

Simplify each term.

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

$f (- 3) = - 2 \cdot 9 - 3 \cdot - 3 - 1$

Multiply $- 2$ by $9$ .

$f (- 3) = - 18 - 3 \cdot - 3 - 1$

Multiply $- 3$ by $- 3$ .

$f (- 3) = - 18 + 9 - 1$

Simplify by adding and subtracting.

Add $- 18$ and $9$ .

$f (- 3) = - 9 - 1$

Subtract $1$ from $- 9$ .

$f (- 3) = - 10$

The final answer is $- 10$ .

$- 10$

The $y$ value at $x = - 3$ is $- 10$ .

$y = - 10$

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

$f (0) = - 2 {(0)}^{2} - 3 \cdot 0 - 1$

Simplify the result.

Simplify each term.

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

$f (0) = - 2 \cdot 0 - 3 \cdot 0 - 1$

Multiply $- 2$ by $0$ .

$f (0) = 0 - 3 \cdot 0 - 1$

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 1$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 1$

Subtract $1$ from $0$ .

$f (0) = - 1$

The final answer is $- 1$ .

$- 1$

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

$y = - 1$

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

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

Simplify the result.

Simplify each term.

One to any power is one.

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

Multiply $- 2$ by $1$ .

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

Multiply $- 3$ by $1$ .

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

Simplify by subtracting numbers.

Subtract $3$ from $- 2$ .

$f (1) = - 5 - 1$

Subtract $1$ from $- 5$ .

$f (1) = - 6$

The final answer is $- 6$ .

$- 6$

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

$y = - 6$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 10 \\ - 2 & - 3 \\ - \frac{3}{4} & \frac{1}{8} \\ 0 & - 1 \\ 1 & - 6 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{3}{4}, \frac{1}{8})$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $x = - \frac{3}{4}$

Directrix: $y = \frac{1}{4}$

$\begin{array}{c} x & y \\ - 3 & - 10 \\ - 2 & - 3 \\ - \frac{3}{4} & \frac{1}{8} \\ 0 & - 1 \\ 1 & - 6 \end{array}$

Step 5

Graph 2x^2+3x+y+1=0

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