Graph x^2

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Algebra

Graph x^2

$x^{2}$

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

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

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

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Factor $2$ out of $0$ .

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

Cancel the common factors.

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Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{0}{1}$

Divide $0$ by $1$ .

$d = 0$

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 positive, the parabola opens up.

Opens Up

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 $1$ .

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Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

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 Up

Vertex: $(0, 0)$

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

Axis of Symmetry: $x = 0$

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

Direction: Opens Up

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

$f (- 1) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

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

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

Simplify the result.

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

$f (- 2) = 4$

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.

$f (1) = 1$

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 $2$ .

$f (2) = 4$

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 Up

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	three hundred ninety-nine million one hundred thirteen thousand sixty-six

Interesting facts

399113066 has 16 divisors, whose sum is 681490368
The reverse of 399113066 is 660311993
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 38
Digital Root 2

Name

Name	one billion nine hundred eighty-one million seventy-one thousand four hundred eighty-eight

Interesting facts

1981071488 has 512 divisors, whose sum is 36925622928
The reverse of 1981071488 is 8841701891
Previous prime number is 11

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 47
Digital Root 2

Name

Name	one billion two hundred fifty-eight million eight hundred sixty-two thousand eight hundred thirty

Interesting facts

1258862830 has 8 divisors, whose sum is 2265953112
The reverse of 1258862830 is 0382688521
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 43
Digital Root 7

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