Graph f(x)=-2x^2-4x+6

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} - 4 x + 6$ .

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

$a = - 2$

$b = - 4$

$c = 6$

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

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

Simplify the right side.

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

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = 6 - \frac{{(- 4)}^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

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

Rewrite $- 4$ as $- 1 (4)$ .

$e = 6 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 2}$

Apply the product rule to $- 1 (4)$ .

$e = 6 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 2}$

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

$e = 6 - \frac{1 \cdot 4^{2}}{4 \cdot - 2}$

Multiply $4^{2}$ by $1$ .

$e = 6 - \frac{4^{2}}{4 \cdot - 2}$

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

$e = 6 - \frac{4 \cdot 4}{4 \cdot - 2}$

Cancel the common factors.

Factor $4$ out of $4 \cdot - 2$ .

$e = 6 - \frac{4 \cdot 4}{4 (- 2)}$

Cancel the common factor.

$e = 6 - \frac{4 \cdot 4}{4 \cdot - 2}$

Rewrite the expression.

$e = 6 - \frac{4}{- 2}$

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

Factor $2$ out of $4$ .

$e = 6 - \frac{2 (2)}{- 2}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$e = 6 - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$e = 6 - - 2$

Multiply $- 1$ by $- 2$ .

$e = 6 + 2$

Add $6$ and $2$ .

$e = 8$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + 1)}^{2} + 8$ .

$- 2 {(x + 1)}^{2} + 8$

Set $y$ equal to the new right side.

$y = - 2 {(x + 1)}^{2} + 8$

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

$a = - 2$

$h = - 1$

$k = 8$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(- 1, 8)$

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.

$\frac{1}{4 \cdot - 2}$

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- 1, \frac{63}{8})$

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.

$y = \frac{65}{8}$

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

Direction: Opens Down

Vertex: $(- 1, 8)$

Focus: $(- 1, \frac{63}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{65}{8}$

Direction: Opens Down

Vertex: $(- 1, 8)$

Focus: $(- 1, \frac{63}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{65}{8}$

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 $- 2$ in the expression.

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

Simplify the result.

Simplify each term.

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

Multiply $- 2$ by ${(- 2)}^{2}$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 2) = {(- 2)}^{1 + 2} - 4 \cdot - 2 + 6$

Add $1$ and $2$ .

$f (- 2) = {(- 2)}^{3} - 4 \cdot - 2 + 6$

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

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

Multiply $- 4$ by $- 2$ .

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

Simplify by adding numbers.

Add $- 8$ and $8$ .

$f (- 2) = 0 + 6$

Add $0$ and $6$ .

$f (- 2) = 6$

The final answer is $6$ .

$6$

The $y$ value at $x = - 2$ is $6$ .

$y = 6$

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

$f (- 3) = - 2 {(- 3)}^{2} - 4 \cdot - 3 + 6$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 2 \cdot 9 - 4 \cdot - 3 + 6$

Multiply $- 2$ by $9$ .

$f (- 3) = - 18 - 4 \cdot - 3 + 6$

Multiply $- 4$ by $- 3$ .

$f (- 3) = - 18 + 12 + 6$

Simplify by adding numbers.

Add $- 18$ and $12$ .

$f (- 3) = - 6 + 6$

Add $- 6$ and $6$ .

$f (- 3) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 3$ is $0$ .

$y = 0$

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

$f (0) = - 2 {(0)}^{2} - 4 \cdot 0 + 6$

Simplify the result.

Simplify each term.

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

$f (0) = - 2 \cdot 0 - 4 \cdot 0 + 6$

Multiply $- 2$ by $0$ .

$f (0) = 0 - 4 \cdot 0 + 6$

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0 + 6$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 6$

Add $0$ and $6$ .

$f (0) = 6$

The final answer is $6$ .

$6$

The $y$ value at $x = 0$ is $6$ .

$y = 6$

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

$f (1) = - 2 {(1)}^{2} - 4 \cdot 1 + 6$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = - 2 \cdot 1 - 4 \cdot 1 + 6$

Multiply $- 2$ by $1$ .

$f (1) = - 2 - 4 \cdot 1 + 6$

Multiply $- 4$ by $1$ .

$f (1) = - 2 - 4 + 6$

Simplify by adding and subtracting.

Subtract $4$ from $- 2$ .

$f (1) = - 6 + 6$

Add $- 6$ and $6$ .

$f (1) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 6 \\ - 1 & 8 \\ 0 & 6 \\ 1 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, 8)$

Focus: $(- 1, \frac{63}{8})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{65}{8}$

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 6 \\ - 1 & 8 \\ 0 & 6 \\ 1 & 0 \end{array}$

Step 4

Graph f(x)=-2x^2-4x+6

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