Graph P(x)=-2(x-3)(x-11)

$P (x) = - 2 (x - 3) (x - 11)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 2 (x - 3) (x - 11)$ .

Simplify the expression.

Apply the distributive property.

$(- 2 x - 2 \cdot - 3) (x - 11)$

Multiply $- 2$ by $- 3$ .

$(- 2 x + 6) (x - 11)$

Expand $(- 2 x + 6) (x - 11)$ using the FOIL Method.

Apply the distributive property.

$- 2 x (x - 11) + 6 (x - 11)$

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 11 + 6 (x - 11)$

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 11 + 6 x + 6 \cdot - 11$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$- 2 (x \cdot x) - 2 x \cdot - 11 + 6 x + 6 \cdot - 11$

Multiply $x$ by $x$ .

$- 2 x^{2} - 2 x \cdot - 11 + 6 x + 6 \cdot - 11$

Multiply $- 11$ by $- 2$ .

$- 2 x^{2} + 22 x + 6 x + 6 \cdot - 11$

Multiply $6$ by $- 11$ .

$- 2 x^{2} + 22 x + 6 x - 66$

Add $22 x$ and $6 x$ .

$- 2 x^{2} + 28 x - 66$

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

$a = - 2$

$b = 28$

$c = - 66$

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

$d = \frac{28}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $28$ and $2$ .

Factor $2$ out of $28$ .

$d = \frac{2 \cdot 14}{2 \cdot - 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot 14}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot 14}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{14}{- 2}$

Cancel the common factor of $14$ and $- 2$ .

Factor $2$ out of $14$ .

$d = \frac{2 (7)}{- 2}$

Move the negative one from the denominator of $\frac{7}{- 1}$ .

$d = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$d = - 7$

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

$e = - 66 - \frac{28^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $28$ to the power of $2$ .

$e = - 66 - \frac{784}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 66 - \frac{784}{- 8}$

Divide $784$ by $- 8$ .

$e = - 66 - - 98$

Multiply $- 1$ by $- 98$ .

$e = - 66 + 98$

Add $- 66$ and $98$ .

$e = 32$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - 7)}^{2} + 32$ .

$- 2 {(x - 7)}^{2} + 32$

Set $y$ equal to the new right side.

$y = - 2 {(x - 7)}^{2} + 32$

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

$a = - 2$

$h = 7$

$k = 32$

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

Opens Down

Find the vertex $(h, k)$ .

$(7, 32)$

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.

$(7, \frac{255}{8})$

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

$x = 7$

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.

$y = \frac{257}{8}$

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

Direction: Opens Down

Vertex: $(7, 32)$

Focus: $(7, \frac{255}{8})$

Axis of Symmetry: $x = 7$

Directrix: $y = \frac{257}{8}$

Direction: Opens Down

Vertex: $(7, 32)$

Focus: $(7, \frac{255}{8})$

Axis of Symmetry: $x = 7$

Directrix: $y = \frac{257}{8}$

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 $6$ in the expression.

$f (6) = - 2 ((6) - 3) ((6) - 11)$

Simplify the result.

Subtract $3$ from $6$ .

$f (6) = - 2 \cdot (3 ((6) - 11))$

Multiply $- 2$ by $3$ .

$f (6) = - 6 ((6) - 11)$

Subtract $11$ from $6$ .

$f (6) = - 6 \cdot - 5$

Multiply $- 6$ by $- 5$ .

$f (6) = 30$

The final answer is $30$ .

$30$

The $y$ value at $x = 6$ is $30$ .

$y = 30$

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 2 ((5) - 3) ((5) - 11)$

Simplify the result.

Subtract $3$ from $5$ .

$f (5) = - 2 \cdot (2 ((5) - 11))$

Multiply $- 2$ by $2$ .

$f (5) = - 4 ((5) - 11)$

Subtract $11$ from $5$ .

$f (5) = - 4 \cdot - 6$

Multiply $- 4$ by $- 6$ .

$f (5) = 24$

The final answer is $24$ .

$24$

The $y$ value at $x = 5$ is $24$ .

$y = 24$

Replace the variable $x$ with $8$ in the expression.

$f (8) = - 2 ((8) - 3) ((8) - 11)$

Simplify the result.

Subtract $3$ from $8$ .

$f (8) = - 2 \cdot (5 ((8) - 11))$

Multiply $- 2$ by $5$ .

$f (8) = - 10 ((8) - 11)$

Subtract $11$ from $8$ .

$f (8) = - 10 \cdot - 3$

Multiply $- 10$ by $- 3$ .

$f (8) = 30$

The final answer is $30$ .

$30$

The $y$ value at $x = 8$ is $30$ .

$y = 30$

Replace the variable $x$ with $9$ in the expression.

$f (9) = - 2 ((9) - 3) ((9) - 11)$

Simplify the result.

Subtract $3$ from $9$ .

$f (9) = - 2 \cdot (6 ((9) - 11))$

Multiply $- 2$ by $6$ .

$f (9) = - 12 ((9) - 11)$

Subtract $11$ from $9$ .

$f (9) = - 12 \cdot - 2$

Multiply $- 12$ by $- 2$ .

$f (9) = 24$

The final answer is $24$ .

$24$

The $y$ value at $x = 9$ is $24$ .

$y = 24$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 5 & 24 \\ 6 & 30 \\ 7 & 32 \\ 8 & 30 \\ 9 & 24 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(7, 32)$

Focus: $(7, \frac{255}{8})$

Axis of Symmetry: $x = 7$

Directrix: $y = \frac{257}{8}$

$\begin{array}{c} x & y \\ 5 & 24 \\ 6 & 30 \\ 7 & 32 \\ 8 & 30 \\ 9 & 24 \end{array}$

Step 4

Graph P(x)=-2(x-3)(x-11)

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