Graph -x^2-3x-1

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Trigonometry

Graph -x^2-3x-1

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

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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

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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.

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Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

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

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

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Simplify each term.

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

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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.

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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.

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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 $- 1$ .

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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.

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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.

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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.

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

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

Simplify the result.

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Simplify each term.

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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.

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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.

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Simplify each term.

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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.

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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.

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Simplify each term.

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

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

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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.

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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.

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Simplify each term.

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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.

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

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

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Name

Name	one billion four hundred thirty-six million eighty-two thousand seven hundred twenty-eight

Interesting facts

1436082728 has 32 divisors, whose sum is 4964994468
The reverse of 1436082728 is 8272806341
Previous prime number is 41

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	one billion six hundred twenty-four million seven hundred twelve thousand nine hundred seventy-one

Interesting facts

1624712971 has 16 divisors, whose sum is 1958376960
The reverse of 1624712971 is 1792174261
Previous prime number is 521

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 40
Digital Root 4

Name

Name	one billion four hundred ten million eight hundred ninety-four thousand two hundred one

Interesting facts

1410894201 has 8 divisors, whose sum is 1881489600
The reverse of 1410894201 is 1024980141
Previous prime number is 6983

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 30
Digital Root 3

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