Graph y=x^2+4x-4

$y = x^{2} + 4 x - 4$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $x^{2} + 4 x - 4$ .

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

$a = 1$

$b = 4$

$c = - 4$

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 $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot 1}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = 2$

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 $e = c - \frac{b^{2}}{4 a}$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = - 4 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = - 4 - 4$

Subtract $4$ from $- 4$ .

$e = - 8$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 2)}^{2} - 8$ .

${(x + 2)}^{2} - 8$

Set $y$ equal to the new right side.

$y = {(x + 2)}^{2} - 8$

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

$a = 1$

$h = - 2$

$k = - 8$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 2, - 8)$

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.

$(- 2, - \frac{31}{4})$

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

$x = - 2$

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.

$y = - \frac{33}{4}$

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

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{31}{4})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{33}{4}$

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{31}{4})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{33}{4}$

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 $- 3$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (- 3) = 9 + 4 (- 3) - 4$

Multiply $4$ by $- 3$ .

$f (- 3) = 9 - 12 - 4$

Simplify by subtracting numbers.

Subtract $12$ from $9$ .

$f (- 3) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (- 3) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = - 3$ is $- 7$ .

$y = - 7$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $4$ by $- 4$ .

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

Simplify by subtracting numbers.

Subtract $16$ from $16$ .

$f (- 4) = 0 - 4$

Subtract $4$ from $0$ .

$f (- 4) = - 4$

The final answer is $- 4$ .

$- 4$

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

$y = - 4$

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

$f (- 1) = {(- 1)}^{2} + 4 (- 1) - 4$

Simplify the result.

Simplify each term.

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

$f (- 1) = 1 + 4 (- 1) - 4$

Multiply $4$ by $- 1$ .

$f (- 1) = 1 - 4 - 4$

Simplify by subtracting numbers.

Subtract $4$ from $1$ .

$f (- 1) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (- 1) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = - 1$ is $- 7$ .

$y = - 7$

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

$f (0) = {(0)}^{2} + 4 (0) - 4$

Simplify the result.

Simplify each term.

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

$f (0) = 0 + 4 (0) - 4$

Multiply $4$ by $0$ .

$f (0) = 0 + 0 - 4$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 4$

Subtract $4$ from $0$ .

$f (0) = - 4$

The final answer is $- 4$ .

$- 4$

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

$y = - 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & - 4 \\ - 3 & - 7 \\ - 2 & - 8 \\ - 1 & - 7 \\ 0 & - 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{31}{4})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{33}{4}$

$\begin{array}{c} x & y \\ - 4 & - 4 \\ - 3 & - 7 \\ - 2 & - 8 \\ - 1 & - 7 \\ 0 & - 4 \end{array}$

Step 4

Graph y=x^2+4x-4

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