Graph y=-x^2

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Algebra

Graph y=-x^2

$y = - x^{2}$

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 $- x^{2}$ .

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

$a = - 1, b = 0, c = 0$

Consider the vertex form of a parabola.

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

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

$d = \frac{0}{2 (- 1)}$

Simplify the right side.

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Cancel the common factor of $0$ and $2$ .

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Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- 1)}$

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Simplify the expression.

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Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Multiply $- 1$ by $0$ .

$d = 0$

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

$e = 0 - \frac{0}{4 (- 1)}$

Multiply $4$ by $- 1$ .

$e = 0 - \frac{0}{- 4}$

Divide $0$ by $- 4$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

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

$- x^{2}$

Set $y$ equal to the new right side.

$y = - x^{2}$

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

$a = - 1$

$h = 0$

$k = 0$

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

Opens Down

Find the vertex $(h, k)$ .

$(0, 0)$

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.

$(0, - \frac{1}{4})$

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

$x = 0$

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.

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

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

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{4})$

Axis of Symmetry: $x = 0$

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

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{4})$

Axis of Symmetry: $x = 0$

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

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 $- 1$ in the expression.

$f (- 1) = - {(- 1)}^{2}$

Simplify the result.

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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 $1$ .

$f (- 1) = (- 1) {(- 1)}^{2}$

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

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

Add $1$ and $2$ .

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

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

$f (- 1) = - 1$

The final answer is $- 1$ .

$- 1$

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

$y = - 1$

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - {(- 2)}^{2}$

Simplify the result.

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

$f (- 2) = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$f (- 2) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = - 2$ is $- 4$ .

$y = - 4$

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

$f (1) = - {(1)}^{2}$

Simplify the result.

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One to any power is one.

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

Multiply $- 1$ by $1$ .

$f (1) = - 1$

The final answer is $- 1$ .

$- 1$

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

$y = - 1$

Replace the variable $x$ with $2$ in the expression.

$f (2) = - {(2)}^{2}$

Simplify the result.

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

$f (2) = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$f (2) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = 2$ is $- 4$ .

$y = - 4$

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{4})$

Axis of Symmetry: $x = 0$

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

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

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Name

Name	five hundred ninety-four million nine hundred eighty-eight thousand eight hundred ninety-five

Interesting facts

594988895 has 16 divisors, whose sum is 761088000
The reverse of 594988895 is 598889495
Previous prime number is 79

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 65
Digital Root 2

Name

Name	one billion nine hundred sixty-two million five hundred eighty-eight thousand thirty-six

Interesting facts

1962588036 has 32 divisors, whose sum is 5987558880
The reverse of 1962588036 is 6308852691
Previous prime number is 59

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 48
Digital Root 3

Name

Name	one billion four hundred sixty-five million one hundred seventy-one thousand one hundred twenty-one

Interesting facts

1465171121 has 8 divisors, whose sum is 1535813664
The reverse of 1465171121 is 1211715641
Previous prime number is 41

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 29
Digital Root 2

Factor y^2-4

Evaluate (-625)^(1/4)

Solve for x square root of 2x+12=x-6

Find the Discriminant -7x^2+8x+2

Convert to a Decimal 57/220