Graph g(x)=-6(x-2)(x)

$g (x) = - 6 (x - 2) (x)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Remove parentheses.

$y = - 6 (x - 2) x$

Simplify $- 6 (x - 2) x$ .

Apply the distributive property.

$y = (- 6 x - 6 \cdot - 2) x$

Multiply $- 6$ by $- 2$ .

$y = (- 6 x + 12) x$

Apply the distributive property.

$y = - 6 x \cdot x + 12 x$

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

Move $x$ .

$y = - 6 (x \cdot x) + 12 x$

Multiply $x$ by $x$ .

$y = - 6 x^{2} + 12 x$

Complete the square for $- 6 x^{2} + 12 x$ .

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

$a = - 6$

$b = 12$

$c = 0$

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

$d = \frac{12}{2 \cdot - 6}$

Simplify the right side.

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot - 6}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 6$ .

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

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot - 6}$

Rewrite the expression.

$d = \frac{6}{- 6}$

Cancel the common factor of $6$ and $- 6$ .

Factor $6$ out of $6$ .

$d = \frac{6 (1)}{- 6}$

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

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{12^{2}}{4 \cdot - 6}$

Simplify the right side.

Simplify each term.

Raise $12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot - 6}$

Multiply $4$ by $- 6$ .

$e = 0 - \frac{144}{- 24}$

Divide $144$ by $- 24$ .

$e = 0 - - 6$

Multiply $- 1$ by $- 6$ .

$e = 0 + 6$

Add $0$ and $6$ .

$e = 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x - 1)}^{2} + 6$ .

$- 6 {(x - 1)}^{2} + 6$

Set $y$ equal to the new right side.

$y = - 6 {(x - 1)}^{2} + 6$

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

$a = - 6$

$h = 1$

$k = 6$

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

Opens Down

Find the vertex $(h, k)$ .

$(1, 6)$

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.

$\frac{1}{4 \cdot - 6}$

Simplify.

Multiply $4$ by $- 6$ .

$\frac{1}{- 24}$

Move the negative in front of the fraction.

$- \frac{1}{24}$

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.

$(1, \frac{143}{24})$

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.

$y = \frac{145}{24}$

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

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{143}{24})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{145}{24}$

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{143}{24})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{145}{24}$

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.

$f (0) = - 6 {(0)}^{2} + 12 (0)$

Simplify the result.

Simplify each term.

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

$f (0) = - 6 \cdot 0 + 12 (0)$

Multiply $- 6$ by $0$ .

$f (0) = 0 + 12 (0)$

Multiply $12$ 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.

$f (- 1) = - 6 {(- 1)}^{2} + 12 (- 1)$

Simplify the result.

Simplify each term.

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

$f (- 1) = - 6 \cdot 1 + 12 (- 1)$

Multiply $- 6$ by $1$ .

$f (- 1) = - 6 + 12 (- 1)$

Multiply $12$ by $- 1$ .

$f (- 1) = - 6 - 12$

Subtract $12$ from $- 6$ .

$f (- 1) = - 18$

The final answer is $- 18$ .

$- 18$

The $y$ value at $x = - 1$ is $- 18$ .

$y = - 18$

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

$f (2) = - 6 {(2)}^{2} + 12 (2)$

Simplify the result.

Simplify each term.

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

$f (2) = - 6 \cdot 4 + 12 (2)$

Multiply $- 6$ by $4$ .

$f (2) = - 24 + 12 (2)$

Multiply $12$ by $2$ .

$f (2) = - 24 + 24$

Add $- 24$ and $24$ .

$f (2) = 0$

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.

$f (3) = - 6 {(3)}^{2} + 12 (3)$

Simplify the result.

Simplify each term.

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

$f (3) = - 6 \cdot 9 + 12 (3)$

Multiply $- 6$ by $9$ .

$f (3) = - 54 + 12 (3)$

Multiply $12$ by $3$ .

$f (3) = - 54 + 36$

Add $- 54$ and $36$ .

$f (3) = - 18$

The final answer is $- 18$ .

$- 18$

The $y$ value at $x = 3$ is $- 18$ .

$y = - 18$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 18 \\ 0 & 0 \\ 1 & 6 \\ 2 & 0 \\ 3 & - 18 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, 6)$

Focus: $(1, \frac{143}{24})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{145}{24}$

$\begin{array}{c} x & y \\ - 1 & - 18 \\ 0 & 0 \\ 1 & 6 \\ 2 & 0 \\ 3 & - 18 \end{array}$

Step 4

Graph g(x)=-6(x-2)(x)

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