Graph y=x^2+3

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Algebra

Graph y=x^2+3

$y = x^{2} + 3$

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 $x^{2} + 3$ .

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

$a = 1, b = 0, c = 3$

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 $0$ .

$e = 3 - \frac{0}{4 (1)}$

Multiply $4$ by $1$ .

$e = 3 - \frac{0}{4}$

Divide $0$ by $4$ .

$e = 3 - 0$

Multiply $- 1$ by $0$ .

$e = 3 + 0$

Add $3$ and $0$ .

$e = 3$

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

${(x + 0)}^{2} + 3$

Set $y$ equal to the new right side.

$y = {(x + 0)}^{2} + 3$

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

$a = 1$

$h = 0$

$k = 3$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(0, 3)$

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.

$(0, \frac{13}{4})$

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.

$y = \frac{11}{4}$

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

Direction: Opens Up

Vertex: $(0, 3)$

Focus: $(0, \frac{13}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{11}{4}$

Direction: Opens Up

Vertex: $(0, 3)$

Focus: $(0, \frac{13}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{11}{4}$

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.

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

Simplify the result.

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

$f (- 1) = 1 + 3$

Add $1$ and $3$ .

$f (- 1) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = - 1$ is $4$ .

$y = 4$

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

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

Simplify the result.

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

$f (- 2) = 4 + 3$

Add $4$ and $3$ .

$f (- 2) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = - 2$ is $7$ .

$y = 7$

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

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

Simplify the result.

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One to any power is one.

$f (1) = 1 + 3$

Add $1$ and $3$ .

$f (1) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 1$ is $4$ .

$y = 4$

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

$f (2) = {(2)}^{2} + 3$

Simplify the result.

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

$f (2) = 4 + 3$

Add $4$ and $3$ .

$f (2) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = 2$ is $7$ .

$y = 7$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 7 \\ - 1 & 4 \\ 0 & 3 \\ 1 & 4 \\ 2 & 7 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(0, 3)$

Focus: $(0, \frac{13}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{11}{4}$

$\begin{array}{c} x & y \\ - 2 & 7 \\ - 1 & 4 \\ 0 & 3 \\ 1 & 4 \\ 2 & 7 \end{array}$

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Name

Name	seventy-six million two hundred eighty thousand nine hundred eleven

Interesting facts

76280911 has 4 divisors, whose sum is 87178192
The reverse of 76280911 is 11908267
Previous prime number is 7

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 34
Digital Root 7

Name

Name	one hundred ninety-nine million thirty-seven thousand eight hundred eighty-nine

Interesting facts

199037889 has 16 divisors, whose sum is 357494592
The reverse of 199037889 is 988730991
Previous prime number is 97

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 54
Digital Root 9

Name

Name	one billion two hundred three million five hundred thousand two hundred seventy-six

Interesting facts

1203500276 has 32 divisors, whose sum is 2955179808
The reverse of 1203500276 is 6720053021
Previous prime number is 5683

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 26
Digital Root 8

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