Graph f(x)=(x-1)^2-1

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

Step 1

Find the properties of the given parabola.

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

$a = 1$

$h = 1$

$k = - 1$

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

Opens Up

Find the vertex $(h, k)$ .

$(1, - 1)$

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.

$\frac{1}{4}$

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.

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

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

$x = 1$

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.

$y = - \frac{5}{4}$

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

Direction: Opens Up

Vertex: $(1, - 1)$

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

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{5}{4}$

Direction: Opens Up

Vertex: $(1, - 1)$

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

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{5}{4}$

Step 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.

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2$ by $- 1$ .

$f (- 1) = 1 + 2$

Add $1$ and $2$ .

$f (- 1) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = - 1$ is $3$ .

$y = 3$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2$ by $2$ .

$f (2) = 4 - 4$

Subtract $4$ from $4$ .

$f (2) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2$ by $3$ .

$f (3) = 9 - 6$

Subtract $6$ from $9$ .

$f (3) = 3$

The final answer is $3$ .

$3$

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

$y = 3$

Graph the parabola using its properties and the selected points.

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, - 1)$

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

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{5}{4}$

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

Step 4

Graph f(x)=(x-1)^2-1

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