Graph -x^2-3x-1

Solve Math

Trigonometry

Graph -x^2-3x-1

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

Find the properties of the given parabola.

Tap for more steps...

Rewrite the equation in vertex form.

Tap for more steps...

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

Tap for more steps...

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

$a = - 1, b = - 3, c = - 1$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Simplify the right side.

Tap for more steps...

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{3}{2}$ .

Tap for more steps...

Multiply $- 1$ by $- 1$ .

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

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

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

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Simplify each term.

Tap for more steps...

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

$e = - 1 - \frac{9}{4 (- 1)}$

Multiply $4$ by $- 1$ .

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

Move the negative in front of the fraction.

$e = - 1 + \frac{9}{4}$

Multiply $- - \frac{9}{4}$ .

Tap for more steps...

Multiply $- 1$ by $- 1$ .

$e = - 1 + 1 (\frac{9}{4})$

Multiply $\frac{9}{4}$ by $1$ .

$e = - 1 + \frac{9}{4}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 1$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Tap for more steps...

Multiply $- 1$ by $4$ .

$e = \frac{- 4 + 9}{4}$

Add $- 4$ and $9$ .

$e = \frac{5}{4}$

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

$- {(x + \frac{3}{2})}^{2} + \frac{5}{4}$

Set $y$ equal to the new right side.

$y = - {(x + \frac{3}{2})}^{2} + \frac{5}{4}$

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

$a = - 1$

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

$k = \frac{5}{4}$

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

Opens Down

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, \frac{5}{4})$

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

Tap for more steps...

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.

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

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

Tap for more steps...

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

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

Find the focus.

Tap for more steps...

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.

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

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

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

Find the directrix.

Tap for more steps...

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.

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

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

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{5}{4})$

Focus: $(- \frac{3}{2}, 1)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{3}{2}$

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{5}{4})$

Focus: $(- \frac{3}{2}, 1)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{3}{2}$

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.

Tap for more steps...

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

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

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

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

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

Multiply $- 1$ by $9$ .

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

Multiply $- 3$ by $- 3$ .

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

Simplify by adding and subtracting.

Tap for more steps...

Add $- 9$ and $9$ .

$f (- 3) = 0 - 1$

Subtract $1$ from $0$ .

$f (- 3) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = - 3$ is $- 1$ .

$y = - 1$

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

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

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

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

$f (- 4) = - 1 \cdot 16 - 3 \cdot - 4 - 1$

Multiply $- 1$ by $16$ .

$f (- 4) = - 16 - 3 \cdot - 4 - 1$

Multiply $- 3$ by $- 4$ .

$f (- 4) = - 16 + 12 - 1$

Simplify by adding and subtracting.

Tap for more steps...

Add $- 16$ and $12$ .

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

Subtract $1$ from $- 4$ .

$f (- 4) = - 5$

The final answer is $- 5$ .

$- 5$

The $y$ value at $x = - 4$ is $- 5$ .

$y = - 5$

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

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

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Multiply $- 1$ by ${(- 1)}^{2}$ .

Tap for more steps...

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

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

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

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

Add $1$ and $2$ .

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

Raise $- 1$ to the power of $3$ .

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

Multiply $- 3$ by $- 1$ .

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

Simplify by adding and subtracting.

Tap for more steps...

Add $- 1$ and $3$ .

$f (- 1) = 2 - 1$

Subtract $1$ from $2$ .

$f (- 1) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

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

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

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

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

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

Multiply $- 1$ by $0$ .

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

Multiply $- 3$ by $0$ .

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

Simplify by adding and subtracting.

Tap for more steps...

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 $- 1$ .

$y = - 1$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & - 5 \\ - 3 & - 1 \\ - \frac{3}{2} & \frac{5}{4} \\ - 1 & 1 \\ 0 & - 1 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{5}{4})$

Focus: $(- \frac{3}{2}, 1)$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{3}{2}$

$\begin{array}{c} x & y \\ - 4 & - 5 \\ - 3 & - 1 \\ - \frac{3}{2} & \frac{5}{4} \\ - 1 & 1 \\ 0 & - 1 \end{array}$

Do you know how to Graph -x^2-3x-1? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name

Name	one billion thirty-six million one hundred thousand one hundred thirty-three

Interesting facts

1036100133 has 16 divisors, whose sum is 1084554240
The reverse of 1036100133 is 3310016301
Previous prime number is 307

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 18
Digital Root 9

Name

Name	one billion one hundred forty-six million five hundred sixty-three thousand seven hundred fifty-seven

Interesting facts

1146563757 has 8 divisors, whose sum is 2038335584
The reverse of 1146563757 is 7573656411
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 45
Digital Root 9

Name

Name	one billion nine hundred twenty-five million thirty-nine thousand six hundred fifty-five

Interesting facts

1925039655 has 8 divisors, whose sum is 2102276880
The reverse of 1925039655 is 5569305291
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 45
Digital Root 9

Find the Inverse y=-3x^3

Simplify tan(270)

Solve for x a=-8+(12-1)+5.25

Solve Graphically k=( natural log of 2)/5750

Expand log of 14