Graph f(x)=8x^2+16x-7

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Trigonometry

$f (x) = 8 x^{2} + 16 x - 7$

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 $8 x^{2} + 16 x - 7$ .

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

$a = 8, b = 16, c = - 7$

Consider the vertex form of a parabola.

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

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

$d = \frac{16}{2 (8)}$

Simplify the right side.

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Cancel the common factor of $16$ and $2$ .

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Factor $2$ out of $16$ .

$d = \frac{2 \cdot 8}{2 \cdot 8}$

Cancel the common factors.

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Factor $2$ out of $2 \cdot 8$ .

$d = \frac{2 \cdot 8}{2 (8)}$

Cancel the common factor.

$d = \frac{2 \cdot 8}{2 \cdot 8}$

Rewrite the expression.

$d = \frac{8}{8}$

Cancel the common factor of $8$ .

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Cancel the common factor.

$d = \frac{8}{8}$

Rewrite the expression.

$d = 1$

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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Raise $16$ to the power of $2$ .

$e = - 7 - \frac{256}{4 (8)}$

Multiply $4$ by $8$ .

$e = - 7 - \frac{256}{32}$

Divide $256$ by $32$ .

$e = - 7 - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$e = - 7 - 8$

Subtract $8$ from $- 7$ .

$e = - 15$

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

$8 {(x + 1)}^{2} - 15$

Set $y$ equal to the new right side.

$y = 8 {(x + 1)}^{2} - 15$

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

$a = 8$

$h = - 1$

$k = - 15$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 1, - 15)$

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.

$\frac{1}{4 \cdot 8}$

Multiply $4$ by $8$ .

$\frac{1}{32}$

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.

$(- 1, - \frac{479}{32})$

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

$x = - 1$

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.

$y = - \frac{481}{32}$

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

Direction: Opens Up

Vertex: $(- 1, - 15)$

Focus: $(- 1, - \frac{479}{32})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{481}{32}$

Direction: Opens Up

Vertex: $(- 1, - 15)$

Focus: $(- 1, - \frac{479}{32})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{481}{32}$

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 $- 2$ in the expression.

$f (- 2) = 8 {(- 2)}^{2} + 16 (- 2) - 7$

Simplify the result.

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Simplify each term.

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

$f (- 2) = 8 \cdot 4 + 16 (- 2) - 7$

Multiply $8$ by $4$ .

$f (- 2) = 32 + 16 (- 2) - 7$

Multiply $16$ by $- 2$ .

$f (- 2) = 32 - 32 - 7$

Simplify by subtracting numbers.

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Subtract $32$ from $32$ .

$f (- 2) = 0 - 7$

Subtract $7$ from $0$ .

$f (- 2) = - 7$

The final answer is $- 7$ .

$- 7$

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

$y = - 7$

Replace the variable $x$ with $0$ in the expression.

$f (0) = 8 {(0)}^{2} + 16 (0) - 7$

Simplify the result.

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Simplify each term.

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

$f (0) = 8 \cdot 0 + 16 (0) - 7$

Multiply $8$ by $0$ .

$f (0) = 0 + 16 (0) - 7$

Multiply $16$ by $0$ .

$f (0) = 0 + 0 - 7$

Simplify by adding and subtracting.

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Add $0$ and $0$ .

$f (0) = 0 - 7$

Subtract $7$ from $0$ .

$f (0) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = 0$ is $- 7$ .

$y = - 7$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - 7 \\ - 1 & - 15 \\ 0 & - 7 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 1, - 15)$

Focus: $(- 1, - \frac{479}{32})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{481}{32}$

$\begin{array}{c} x & y \\ - 2 & - 7 \\ - 1 & - 15 \\ 0 & - 7 \end{array}$

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Name

Name	two billion one hundred twenty-nine million seven hundred twenty-seven thousand five hundred forty-two

Interesting facts

2129727542 has 8 divisors, whose sum is 3202804800
The reverse of 2129727542 is 2457279212
Previous prime number is 389

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	five hundred fifty-four million six hundred forty-two thousand four hundred thirteen

Interesting facts

554642413 has 8 divisors, whose sum is 597542400
The reverse of 554642413 is 314246455
Previous prime number is 3119

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 34
Digital Root 7

Name

Name	one billion eighty-six million eight hundred forty-five thousand five hundred fifty-two

Interesting facts

1086845552 has 64 divisors, whose sum is 5925399480
The reverse of 1086845552 is 2555486801
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 44
Digital Root 8

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