Graph f(x)=x^2-3x-2

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Trigonometry

Graph f(x)=x^2-3x-2

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

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 $x^{2} - 3 x - 2$ .

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

$a = 1, b = - 3, c = - 2$

Consider the vertex form of a parabola.

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

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

$d = - \frac{3}{2 (1)}$

Multiply $2$ by $1$ .

$d = - \frac{3}{2}$

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 $- 3$ to the power of $2$ .

$e = - 2 - \frac{9}{4 (1)}$

Multiply $4$ by $1$ .

$e = - 2 - \frac{9}{4}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 2 \cdot \frac{4}{4} - \frac{9}{4}$

Combine $- 2$ and $\frac{4}{4}$ .

$e = \frac{- 2 \cdot 4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 4 - 9}{4}$

Simplify the numerator.

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Multiply $- 2$ by $4$ .

$e = \frac{- 8 - 9}{4}$

Subtract $9$ from $- 8$ .

$e = \frac{- 17}{4}$

Move the negative in front of the fraction.

$e = - \frac{17}{4}$

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

${(x - \frac{3}{2})}^{2} - \frac{17}{4}$

Set $y$ equal to the new right side.

$y = {(x - \frac{3}{2})}^{2} - \frac{17}{4}$

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

$a = 1$

$h = \frac{3}{2}$

$k = - \frac{17}{4}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{3}{2}, - \frac{17}{4})$

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.

$(\frac{3}{2}, - 4)$

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

$x = \frac{3}{2}$

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.

$y = - \frac{9}{2}$

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

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

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 $1$ in the expression.

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

Simplify the result.

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Simplify each term.

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One to any power is one.

$f (1) = 1 - 3 \cdot 1 - 2$

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 - 2$

Simplify by subtracting numbers.

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Subtract $3$ from $1$ .

$f (1) = - 2 - 2$

Subtract $2$ from $- 2$ .

$f (1) = - 4$

The final answer is $- 4$ .

$- 4$

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

$y = - 4$

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

$f (0) = {(0)}^{2} - 3 \cdot 0 - 2$

Simplify the result.

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Simplify each term.

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

$f (0) = 0 - 3 \cdot 0 - 2$

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 - 2$

Simplify by adding and subtracting.

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Add $0$ and $0$ .

$f (0) = 0 - 2$

Subtract $2$ from $0$ .

$f (0) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = 0$ is $- 2$ .

$y = - 2$

Replace the variable $x$ with $3$ in the expression.

$f (3) = {(3)}^{2} - 3 \cdot 3 - 2$

Simplify the result.

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Simplify each term.

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Raise $3$ to the power of $2$ .

$f (3) = 9 - 3 \cdot 3 - 2$

Multiply $- 3$ by $3$ .

$f (3) = 9 - 9 - 2$

Simplify by subtracting numbers.

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Subtract $9$ from $9$ .

$f (3) = 0 - 2$

Subtract $2$ from $0$ .

$f (3) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = 3$ is $- 2$ .

$y = - 2$

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

$f (4) = {(4)}^{2} - 3 \cdot 4 - 2$

Simplify the result.

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Simplify each term.

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

$f (4) = 16 - 3 \cdot 4 - 2$

Multiply $- 3$ by $4$ .

$f (4) = 16 - 12 - 2$

Simplify by subtracting numbers.

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Subtract $12$ from $16$ .

$f (4) = 4 - 2$

Subtract $2$ from $4$ .

$f (4) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = 4$ is $2$ .

$y = 2$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 4 \\ \frac{3}{2} & - \frac{17}{4} \\ 3 & - 2 \\ 4 & 2 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{17}{4})$

Focus: $(\frac{3}{2}, - 4)$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{9}{2}$

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 4 \\ \frac{3}{2} & - \frac{17}{4} \\ 3 & - 2 \\ 4 & 2 \end{array}$

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Name

Name	one billion nine hundred sixty-one million three hundred twenty-six thousand twenty

Interesting facts

1961326020 has 32 divisors, whose sum is 4821997824
The reverse of 1961326020 is 0206231691
Previous prime number is 41

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 30
Digital Root 3

Name

Name	one billion four hundred thirty-three million eight hundred sixty-three thousand seven hundred fifty-four

Interesting facts

1433863754 has 4 divisors, whose sum is 2150795634
The reverse of 1433863754 is 4573683341
Previous prime number is 2

Basic properties

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

Name

Name	four hundred seventy-four million five hundred ninety-five thousand two hundred fifty-one

Interesting facts

474595251 has 16 divisors, whose sum is 682746624
The reverse of 474595251 is 152595474
Previous prime number is 547

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 42
Digital Root 6

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