Graph f(x)=x^2-11x-26

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Trigonometry

$f (x) = x^{2} - 11 x - 26$

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 $x^{2} - 11 x - 26$ .

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

$a = 1, b = - 11, c = - 26$

Consider the vertex form of a parabola.

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

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

$d = - \frac{11}{2 (1)}$

Multiply $2$ by $1$ .

$d = - \frac{11}{2}$

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 $- 11$ to the power of $2$ .

$e = - 26 - \frac{121}{4 (1)}$

Multiply $4$ by $1$ .

$e = - 26 - \frac{121}{4}$

To write $- 26$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 26 \cdot \frac{4}{4} - \frac{121}{4}$

Combine $- 26$ and $\frac{4}{4}$ .

$e = \frac{- 26 \cdot 4}{4} - \frac{121}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 26 \cdot 4 - 121}{4}$

Simplify the numerator.

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Multiply $- 26$ by $4$ .

$e = \frac{- 104 - 121}{4}$

Subtract $121$ from $- 104$ .

$e = \frac{- 225}{4}$

Move the negative in front of the fraction.

$e = - \frac{225}{4}$

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

${(x - \frac{11}{2})}^{2} - \frac{225}{4}$

Set $y$ equal to the new right side.

$y = {(x - \frac{11}{2})}^{2} - \frac{225}{4}$

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

$a = 1$

$h = \frac{11}{2}$

$k = - \frac{225}{4}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{11}{2}, - \frac{225}{4})$

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.

$(\frac{11}{2}, - 56)$

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

$x = \frac{11}{2}$

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.

$y = - \frac{113}{2}$

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

Direction: Opens Up

Vertex: $(\frac{11}{2}, - \frac{225}{4})$

Focus: $(\frac{11}{2}, - 56)$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = - \frac{113}{2}$

Direction: Opens Up

Vertex: $(\frac{11}{2}, - \frac{225}{4})$

Focus: $(\frac{11}{2}, - 56)$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = - \frac{113}{2}$

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 $5$ in the expression.

$f (5) = {(5)}^{2} - 11 \cdot 5 - 26$

Simplify the result.

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Simplify each term.

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

$f (5) = 25 - 11 \cdot 5 - 26$

Multiply $- 11$ by $5$ .

$f (5) = 25 - 55 - 26$

Simplify by subtracting numbers.

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Subtract $55$ from $25$ .

$f (5) = - 30 - 26$

Subtract $26$ from $- 30$ .

$f (5) = - 56$

The final answer is $- 56$ .

$- 56$

The $y$ value at $x = 5$ is $- 56$ .

$y = - 56$

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 11 \cdot 4 - 26$

Simplify the result.

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Simplify each term.

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

$f (4) = 16 - 11 \cdot 4 - 26$

Multiply $- 11$ by $4$ .

$f (4) = 16 - 44 - 26$

Simplify by subtracting numbers.

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Subtract $44$ from $16$ .

$f (4) = - 28 - 26$

Subtract $26$ from $- 28$ .

$f (4) = - 54$

The final answer is $- 54$ .

$- 54$

The $y$ value at $x = 4$ is $- 54$ .

$y = - 54$

Replace the variable $x$ with $7$ in the expression.

$f (7) = {(7)}^{2} - 11 \cdot 7 - 26$

Simplify the result.

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Simplify each term.

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Raise $7$ to the power of $2$ .

$f (7) = 49 - 11 \cdot 7 - 26$

Multiply $- 11$ by $7$ .

$f (7) = 49 - 77 - 26$

Simplify by subtracting numbers.

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Subtract $77$ from $49$ .

$f (7) = - 28 - 26$

Subtract $26$ from $- 28$ .

$f (7) = - 54$

The final answer is $- 54$ .

$- 54$

The $y$ value at $x = 7$ is $- 54$ .

$y = - 54$

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} - 11 \cdot 8 - 26$

Simplify the result.

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Simplify each term.

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Raise $8$ to the power of $2$ .

$f (8) = 64 - 11 \cdot 8 - 26$

Multiply $- 11$ by $8$ .

$f (8) = 64 - 88 - 26$

Simplify by subtracting numbers.

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Subtract $88$ from $64$ .

$f (8) = - 24 - 26$

Subtract $26$ from $- 24$ .

$f (8) = - 50$

The final answer is $- 50$ .

$- 50$

The $y$ value at $x = 8$ is $- 50$ .

$y = - 50$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & - 54 \\ 5 & - 56 \\ \frac{11}{2} & - \frac{225}{4} \\ 7 & - 54 \\ 8 & - 50 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{11}{2}, - \frac{225}{4})$

Focus: $(\frac{11}{2}, - 56)$

Axis of Symmetry: $x = \frac{11}{2}$

Directrix: $y = - \frac{113}{2}$

$\begin{array}{c} x & y \\ 4 & - 54 \\ 5 & - 56 \\ \frac{11}{2} & - \frac{225}{4} \\ 7 & - 54 \\ 8 & - 50 \end{array}$

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Name

Name	one billion six hundred ninety-one million one hundred twenty thousand eight hundred fifty-seven

Interesting facts

1691120857 has 4 divisors, whose sum is 1692155556
The reverse of 1691120857 is 7580211961
Previous prime number is 1637

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 40
Digital Root 4

Name

Name	two billion one hundred eight million two hundred twenty-eight thousand six hundred eighty-nine

Interesting facts

2108228689 has 16 divisors, whose sum is 2149353792
The reverse of 2108228689 is 9868228012
Previous prime number is 173

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 46
Digital Root 1

Name

Name	two hundred eighty-five million sixty thousand three hundred one

Interesting facts

285060301 has 4 divisors, whose sum is 301828572
The reverse of 285060301 is 103060582
Previous prime number is 17

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 25
Digital Root 7

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