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

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

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	one billion eight hundred seventy-three million five hundred eleven thousand two hundred fifty-four

Interesting facts

1873511254 has 8 divisors, whose sum is 2812568400
The reverse of 1873511254 is 4521153781
Previous prime number is 1223

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 37
Digital Root 1

Name

Name	one billion one hundred seventy-eight million one hundred forty-three thousand five hundred seventy-one

Interesting facts

1178143571 has 8 divisors, whose sum is 1185661104
The reverse of 1178143571 is 1753418711
Previous prime number is 1193

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 38
Digital Root 2

Name

Name	five hundred sixty-eight million three hundred forty-six thousand eight hundred nineteen

Interesting facts

568346819 has 4 divisors, whose sum is 573561120
The reverse of 568346819 is 918643865
Previous prime number is 109

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 50
Digital Root 5

Evaluate (-5)^3

Solve by Factoring x^3+3x^2-4x-12=0

Simplify i^66

Factor x^2-12

Evaluate 20/4