Graph y=(x+3)^2

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Algebra

Graph y=(x+3)^2

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

Find the properties of the given parabola.

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

$a = 1$

$h = - 3$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 3, 0)$

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.

$(- 3, \frac{1}{4})$

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

$x = - 3$

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: $(- 3, 0)$

Focus: $(- 3, \frac{1}{4})$

Axis of Symmetry: $x = - 3$

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

Direction: Opens Up

Vertex: $(- 3, 0)$

Focus: $(- 3, \frac{1}{4})$

Axis of Symmetry: $x = - 3$

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 $- 4$ in the expression.

$f (- 4) = {(- 4)}^{2} + 6 (- 4) + 9$

Simplify the result.

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Simplify each term.

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Raise $- 4$ to the power of $2$ .

$f (- 4) = 16 + 6 (- 4) + 9$

Multiply $6$ by $- 4$ .

$f (- 4) = 16 - 24 + 9$

Simplify by adding and subtracting.

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Subtract $24$ from $16$ .

$f (- 4) = - 8 + 9$

Add $- 8$ and $9$ .

$f (- 4) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = {(- 5)}^{2} + 6 (- 5) + 9$

Simplify the result.

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Simplify each term.

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Raise $- 5$ to the power of $2$ .

$f (- 5) = 25 + 6 (- 5) + 9$

Multiply $6$ by $- 5$ .

$f (- 5) = 25 - 30 + 9$

Simplify by adding and subtracting.

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Subtract $30$ from $25$ .

$f (- 5) = - 5 + 9$

Add $- 5$ and $9$ .

$f (- 5) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = - 5$ is $4$ .

$y = 4$

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

$f (- 2) = {(- 2)}^{2} + 6 (- 2) + 9$

Simplify the result.

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Simplify each term.

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

$f (- 2) = 4 + 6 (- 2) + 9$

Multiply $6$ by $- 2$ .

$f (- 2) = 4 - 12 + 9$

Simplify by adding and subtracting.

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Subtract $12$ from $4$ .

$f (- 2) = - 8 + 9$

Add $- 8$ and $9$ .

$f (- 2) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = - 2$ is $1$ .

$y = 1$

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

$f (- 1) = {(- 1)}^{2} + 6 (- 1) + 9$

Simplify the result.

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Simplify each term.

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

$f (- 1) = 1 + 6 (- 1) + 9$

Multiply $6$ by $- 1$ .

$f (- 1) = 1 - 6 + 9$

Simplify by adding and subtracting.

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Subtract $6$ from $1$ .

$f (- 1) = - 5 + 9$

Add $- 5$ and $9$ .

$f (- 1) = 4$

The final answer is $4$ .

$4$

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

$y = 4$

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, 0)$

Focus: $(- 3, \frac{1}{4})$

Axis of Symmetry: $x = - 3$

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

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

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Name

Name	two hundred twenty-five million eight hundred twenty-nine thousand one hundred seventy-nine

Interesting facts

225829179 has 4 divisors, whose sum is 250921320
The reverse of 225829179 is 971928522
Previous prime number is 9

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 45
Digital Root 9

Name

Name	nine hundred fifty million seven hundred thirty-seven thousand four hundred three

Interesting facts

950737403 has 16 divisors, whose sum is 1186099200
The reverse of 950737403 is 304737059
Previous prime number is 2111

Basic properties

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

Name

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

Interesting facts

1166383239 has 8 divisors, whose sum is 1777345920
The reverse of 1166383239 is 9323836611
Previous prime number is 7

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 42
Digital Root 6

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