Graph y+3x^2=6x-1

$y + 3 x^{2} = 6 x - 1$

Step 1

Subtract $3 x^{2}$ from both sides of the equation.

$y = 6 x - 1 - 3 x^{2}$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify $6 x - 1 - 3 x^{2}$ .

Move $- 1$ .

$y = 6 x - 3 x^{2} - 1$

Reorder $6 x$ and $- 3 x^{2}$ .

$y = - 3 x^{2} + 6 x - 1$

Complete the square for $- 3 x^{2} + 6 x - 1$ .

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

$a = - 3$

$b = 6$

$c = - 1$

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

$d = \frac{6}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $3$ out of $3$ .

$d = \frac{3 (1)}{- 3}$

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

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

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = - 1 - \frac{36}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 1 - \frac{36}{- 12}$

Divide $36$ by $- 12$ .

$e = - 1 - - 3$

Multiply $- 1$ by $- 3$ .

$e = - 1 + 3$

Add $- 1$ and $3$ .

$e = 2$

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

$- 3 {(x - 1)}^{2} + 2$

Set $y$ equal to the new right side.

$y = - 3 {(x - 1)}^{2} + 2$

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

$a = - 3$

$h = 1$

$k = 2$

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

Opens Down

Find the vertex $(h, k)$ .

$(1, 2)$

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.

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

Simplify.

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Move the negative in front of the fraction.

$- \frac{1}{12}$

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.

$(1, \frac{23}{12})$

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.

$y = \frac{25}{12}$

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

Direction: Opens Down

Vertex: $(1, 2)$

Focus: $(1, \frac{23}{12})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{25}{12}$

Direction: Opens Down

Vertex: $(1, 2)$

Focus: $(1, \frac{23}{12})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{25}{12}$

Step 3

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.

$f (0) = - 3 {(0)}^{2} + 6 (0) - 1$

Simplify the result.

Simplify each term.

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

$f (0) = - 3 \cdot 0 + 6 (0) - 1$

Multiply $- 3$ by $0$ .

$f (0) = 0 + 6 (0) - 1$

Multiply $6$ by $0$ .

$f (0) = 0 + 0 - 1$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 1$

Subtract $1$ from $0$ .

$f (0) = - 1$

The final answer is $- 1$ .

$- 1$

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

$y = - 1$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 3$ by $1$ .

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

Multiply $6$ by $- 1$ .

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

Simplify by subtracting numbers.

Subtract $6$ from $- 3$ .

$f (- 1) = - 9 - 1$

Subtract $1$ from $- 9$ .

$f (- 1) = - 10$

The final answer is $- 10$ .

$- 10$

The $y$ value at $x = - 1$ is $- 10$ .

$y = - 10$

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

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

Simplify the result.

Simplify each term.

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

$f (2) = - 3 \cdot 4 + 6 (2) - 1$

Multiply $- 3$ by $4$ .

$f (2) = - 12 + 6 (2) - 1$

Multiply $6$ by $2$ .

$f (2) = - 12 + 12 - 1$

Simplify by adding and subtracting.

Add $- 12$ and $12$ .

$f (2) = 0 - 1$

Subtract $1$ from $0$ .

$f (2) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = 2$ is $- 1$ .

$y = - 1$

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

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

Simplify the result.

Simplify each term.

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

$f (3) = - 3 \cdot 9 + 6 (3) - 1$

Multiply $- 3$ by $9$ .

$f (3) = - 27 + 6 (3) - 1$

Multiply $6$ by $3$ .

$f (3) = - 27 + 18 - 1$

Simplify by adding and subtracting.

Add $- 27$ and $18$ .

$f (3) = - 9 - 1$

Subtract $1$ from $- 9$ .

$f (3) = - 10$

The final answer is $- 10$ .

$- 10$

The $y$ value at $x = 3$ is $- 10$ .

$y = - 10$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 10 \\ 0 & - 1 \\ 1 & 2 \\ 2 & - 1 \\ 3 & - 10 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, 2)$

Focus: $(1, \frac{23}{12})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{25}{12}$

$\begin{array}{c} x & y \\ - 1 & - 10 \\ 0 & - 1 \\ 1 & 2 \\ 2 & - 1 \\ 3 & - 10 \end{array}$

Step 5

Graph y+3x^2=6x-1

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