Graph f(x)=(X+3)^2-1

$f (x) = {(X + 3)}^{2} - 1$

Step 1

The function declaration $f (x)$ varies according to $x$ , but the input function ${(X + 3)}^{2} - 1$ only contains the variable $X$ . Assume $f (X) = {(X + 3)}^{2} - 1$ .

$f (X) = {(X + 3)}^{2} - 1$

Step 2

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 $k$ .

$a = 1$

$h = - 3$

$k = - 1$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 3, - 1)$

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.

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

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

$X = - 3$

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: $(- 3, - 1)$

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

Axis of Symmetry: $X = - 3$

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

Direction: Opens Up

Vertex: $(- 3, - 1)$

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

Axis of Symmetry: $X = - 3$

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

Step 3

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 $- 4$ in the expression.

$f (- 4) = {(- 4)}^{2} + 6 (- 4) + 8$

Simplify the result.

Simplify each term.

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

$f (- 4) = 16 + 6 (- 4) + 8$

Multiply $6$ by $- 4$ .

$f (- 4) = 16 - 24 + 8$

Simplify by adding and subtracting.

Subtract $24$ from $16$ .

$f (- 4) = - 8 + 8$

Add $- 8$ and $8$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

The $y$ value at $X = - 4$ is $0$ .

$y = 0$

Replace the variable $X$ with $- 5$ in the expression.

$f (- 5) = {(- 5)}^{2} + 6 (- 5) + 8$

Simplify the result.

Simplify each term.

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

$f (- 5) = 25 + 6 (- 5) + 8$

Multiply $6$ by $- 5$ .

$f (- 5) = 25 - 30 + 8$

Simplify by adding and subtracting.

Subtract $30$ from $25$ .

$f (- 5) = - 5 + 8$

Add $- 5$ and $8$ .

$f (- 5) = 3$

The final answer is $3$ .

$3$

The $y$ value at $X = - 5$ is $3$ .

$y = 3$

Replace the variable $X$ with $- 2$ in the expression.

$f (- 2) = {(- 2)}^{2} + 6 (- 2) + 8$

Simplify the result.

Simplify each term.

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

$f (- 2) = 4 + 6 (- 2) + 8$

Multiply $6$ by $- 2$ .

$f (- 2) = 4 - 12 + 8$

Simplify by adding and subtracting.

Subtract $12$ from $4$ .

$f (- 2) = - 8 + 8$

Add $- 8$ and $8$ .

$f (- 2) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

$f (- 1) = {(- 1)}^{2} + 6 (- 1) + 8$

Simplify the result.

Simplify each term.

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

$f (- 1) = 1 + 6 (- 1) + 8$

Multiply $6$ by $- 1$ .

$f (- 1) = 1 - 6 + 8$

Simplify by adding and subtracting.

Subtract $6$ from $1$ .

$f (- 1) = - 5 + 8$

Add $- 5$ and $8$ .

$f (- 1) = 3$

The final answer is $3$ .

$3$

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

$y = 3$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & 3 \\ - 4 & 0 \\ - 3 & - 1 \\ - 2 & 0 \\ - 1 & 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, - 1)$

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

Axis of Symmetry: $X = - 3$

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

$\begin{array}{c} x & y \\ - 5 & 3 \\ - 4 & 0 \\ - 3 & - 1 \\ - 2 & 0 \\ - 1 & 3 \end{array}$

Step 5

Graph f(x)=(X+3)^2-1

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