Graph f(x)=3x^2-3x-6

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} - 3 x - 6$ .

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

$a = 3$

$b = - 3$

$c = - 6$

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

$d = \frac{- 3}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3$ .

$d = \frac{3 \cdot - 1}{2 \cdot 3}$

Cancel the common factors.

Factor $3$ out of $2 \cdot 3$ .

$d = \frac{3 \cdot - 1}{3 \cdot 2}$

Cancel the common factor.

$d = \frac{3 \cdot - 1}{3 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$e = - 6 - \frac{{(- 3)}^{2}}{4 \cdot 3}$

Simplify the right side.

Cancel the common factor of ${(- 3)}^{2}$ and $3$ .

Rewrite $- 3$ as $- 1 (3)$ .

$e = - 6 - \frac{{(- 1 (3))}^{2}}{4 \cdot 3}$

Apply the product rule to $- 1 (3)$ .

$e = - 6 - \frac{{(- 1)}^{2} \cdot 3^{2}}{4 \cdot 3}$

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

$e = - 6 - \frac{1 \cdot 3^{2}}{4 \cdot 3}$

Multiply $3^{2}$ by $1$ .

$e = - 6 - \frac{3^{2}}{4 \cdot 3}$

Factor $3$ out of $3^{2}$ .

$e = - 6 - \frac{3 \cdot 3}{4 \cdot 3}$

Cancel the common factors.

Factor $3$ out of $4 \cdot 3$ .

$e = - 6 - \frac{3 \cdot 3}{3 \cdot 4}$

Cancel the common factor.

$e = - 6 - \frac{3 \cdot 3}{3 \cdot 4}$

Rewrite the expression.

$e = - 6 - \frac{3}{4}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 6 \cdot \frac{4}{4} - \frac{3}{4}$

Combine $- 6$ and $\frac{4}{4}$ .

$e = \frac{- 6 \cdot 4}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 4 - 3}{4}$

Simplify the numerator.

Multiply $- 6$ by $4$ .

$e = \frac{- 24 - 3}{4}$

Subtract $3$ from $- 24$ .

$e = \frac{- 27}{4}$

Move the negative in front of the fraction.

$e = - \frac{27}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$ .

$3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{1}{2})}^{2} - \frac{27}{4}$

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

$a = 3$

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

$k = - \frac{27}{4}$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{27}{4})$

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.

$\frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$\frac{1}{12}$

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.

$(\frac{1}{2}, - \frac{20}{3})$

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

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

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.

$y = - \frac{41}{6}$

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

Direction: Opens Up

Vertex: $(\frac{1}{2}, - \frac{27}{4})$

Focus: $(\frac{1}{2}, - \frac{20}{3})$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = - \frac{41}{6}$

Direction: Opens Up

Vertex: $(\frac{1}{2}, - \frac{27}{4})$

Focus: $(\frac{1}{2}, - \frac{20}{3})$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = - \frac{41}{6}$

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 $- 1$ in the expression.

$f (- 1) = 3 {(- 1)}^{2} - 3 \cdot - 1 - 6$

Simplify the result.

Simplify each term.

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

$f (- 1) = 3 \cdot 1 - 3 \cdot - 1 - 6$

Multiply $3$ by $1$ .

$f (- 1) = 3 - 3 \cdot - 1 - 6$

Multiply $- 3$ by $- 1$ .

$f (- 1) = 3 + 3 - 6$

Simplify by adding and subtracting.

Add $3$ and $3$ .

$f (- 1) = 6 - 6$

Subtract $6$ from $6$ .

$f (- 1) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $3$ by $4$ .

$f (- 2) = 12 - 3 \cdot - 2 - 6$

Multiply $- 3$ by $- 2$ .

$f (- 2) = 12 + 6 - 6$

Simplify by adding and subtracting.

Add $12$ and $6$ .

$f (- 2) = 18 - 6$

Subtract $6$ from $18$ .

$f (- 2) = 12$

The final answer is $12$ .

$12$

The $y$ value at $x = - 2$ is $12$ .

$y = 12$

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

$f (1) = 3 {(1)}^{2} - 3 \cdot 1 - 6$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 3 \cdot 1 - 3 \cdot 1 - 6$

Multiply $3$ by $1$ .

$f (1) = 3 - 3 \cdot 1 - 6$

Multiply $- 3$ by $1$ .

$f (1) = 3 - 3 - 6$

Simplify by subtracting numbers.

Subtract $3$ from $3$ .

$f (1) = 0 - 6$

Subtract $6$ from $0$ .

$f (1) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = 1$ is $- 6$ .

$y = - 6$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $3$ by $4$ .

$f (2) = 12 - 3 \cdot 2 - 6$

Multiply $- 3$ by $2$ .

$f (2) = 12 - 6 - 6$

Simplify by subtracting numbers.

Subtract $6$ from $12$ .

$f (2) = 6 - 6$

Subtract $6$ from $6$ .

$f (2) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 2$ is $0$ .

$y = 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 12 \\ - 1 & 0 \\ \frac{1}{2} & - \frac{27}{4} \\ 1 & - 6 \\ 2 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{1}{2}, - \frac{27}{4})$

Focus: $(\frac{1}{2}, - \frac{20}{3})$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = - \frac{41}{6}$

$\begin{array}{c} x & y \\ - 2 & 12 \\ - 1 & 0 \\ \frac{1}{2} & - \frac{27}{4} \\ 1 & - 6 \\ 2 & 0 \end{array}$

Step 4

Graph f(x)=3x^2-3x-6

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