Find the Properties f(x)=x^2-2x-1

$f (x) = x^{2} - 2 x - 1$

Step 1

Write $f (x) = x^{2} - 2 x - 1$ as an equation.

$y = x^{2} - 2 x - 1$

Step 2

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 2 x - 1$ .

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

$a = 1$

$b = - 2$

$c = - 1$

Consider the vertex form of a parabola.

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

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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

$d = \frac{- 2}{2 \cdot 1}$

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

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

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

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

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

$e = - 1 - \frac{4}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 1 - \frac{4}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$e = - 1 - \frac{4}{4}$

Rewrite the expression.

$e = - 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = - 1 - 1$

Subtract $1$ from $- 1$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 1)}^{2} - 2$ .

${(x - 1)}^{2} - 2$

Set $y$ equal to the new right side.

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

Step 3

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

$a = 1$

$h = 1$

$k = - 2$

Step 4

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

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(1, - 2)$

Step 6

Find $p$ , the distance from the vertex to the focus.

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 $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

Step 7

Find the focus.

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.

$(1, - \frac{7}{4})$

Step 8

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

$x = 1$

Step 9

Find the directrix.

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.

$y = - \frac{9}{4}$

Step 10

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

Direction: Opens Up

Vertex: $(1, - 2)$

Focus: $(1, - \frac{7}{4})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{9}{4}$

Step 11

Find the Properties f(x)=x^2-2x-1

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