Graph -1/3x^2+2x-4

$- \frac{1}{3} x^{2} + 2 x - 4$

Step 1

Combine $x^{2}$ and $\frac{1}{3}$ .

$y = - \frac{x^{2}}{3} + 2 x - 4$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{x^{2}}{3} + 2 x - 4$ .

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

$a = - \frac{1}{3}$

$b = 2$

$c = - 4$

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

$d = \frac{2}{2 (- \frac{1}{3})}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 (- \frac{1}{3})}$

Rewrite the expression.

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

Cancel the common factor of $1$ and $- 1$ .

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 3)$

Multiply $- (1 \cdot 3)$ .

Multiply $3$ by $1$ .

$d = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$d = - 3$

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

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

Simplify the right side.

Simplify each term.

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

$e = - 4 - \frac{4}{4 (- \frac{1}{3})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 4 - \frac{4}{- 4 (\frac{1}{3})}$

Combine $- 4$ and $\frac{1}{3}$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$e = - 4 - (4 (- \frac{3}{4}))$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$e = - 4 - (4 (\frac{- 3}{4}))$

Cancel the common factor.

$e = - 4 - (4 (\frac{- 3}{4}))$

Rewrite the expression.

$e = - 4 - - 3$

Multiply $- 1$ by $- 3$ .

$e = - 4 + 3$

Add $- 4$ and $3$ .

$e = - 1$

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

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

Set $y$ equal to the new right side.

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

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

$a = - \frac{1}{3}$

$h = 3$

$k = - 1$

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

Opens Down

Find the vertex $(h, k)$ .

$(3, - 1)$

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.

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

Simplify.

Cancel the common factor of $1$ and $- 1$ .

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

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

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{3})}$

Combine $4$ and $\frac{1}{3}$ .

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

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{4}))$

Multiply $\frac{3}{4}$ by $1$ .

$- \frac{3}{4}$

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.

$(3, - \frac{7}{4})$

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

$x = 3$

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.

$y = - \frac{1}{4}$

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

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - \frac{7}{4})$

Axis of Symmetry: $x = 3$

Directrix: $y = - \frac{1}{4}$

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - \frac{7}{4})$

Axis of Symmetry: $x = 3$

Directrix: $y = - \frac{1}{4}$

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 $2$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (2) = - \frac{4}{3} + 2 (2) - 4$

Multiply $2$ by $2$ .

$f (2) = - \frac{4}{3} + 4 - 4$

Find the common denominator.

Write $4$ as a fraction with denominator $1$ .

$f (2) = - \frac{4}{3} + \frac{4}{1} - 4$

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

$f (2) = - \frac{4}{3} + \frac{4}{1} \cdot \frac{3}{3} - 4$

Multiply $\frac{4}{1}$ by $\frac{3}{3}$ .

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

Write $- 4$ as a fraction with denominator $1$ .

$f (2) = - \frac{4}{3} + \frac{4 \cdot 3}{3} + \frac{- 4}{1}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$f (2) = - \frac{4}{3} + \frac{4 \cdot 3}{3} + \frac{- 4}{1} \cdot \frac{3}{3}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $4$ by $3$ .

$f (2) = \frac{- 4 + 12 - 4 \cdot 3}{3}$

Multiply $- 4$ by $3$ .

$f (2) = \frac{- 4 + 12 - 12}{3}$

Simplify the expression.

Add $- 4$ and $12$ .

$f (2) = \frac{8 - 12}{3}$

Subtract $12$ from $8$ .

$f (2) = \frac{- 4}{3}$

Move the negative in front of the fraction.

$f (2) = - \frac{4}{3}$

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

The $y$ value at $x = 2$ is $- \frac{4}{3}$ .

$y = - \frac{4}{3}$

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

$f (1) = - \frac{{(1)}^{2}}{3} + 2 (1) - 4$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = - \frac{1}{3} + 2 (1) - 4$

Multiply $2$ by $1$ .

$f (1) = - \frac{1}{3} + 2 - 4$

Find the common denominator.

Write $2$ as a fraction with denominator $1$ .

$f (1) = - \frac{1}{3} + \frac{2}{1} - 4$

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (1) = - \frac{1}{3} + \frac{2}{1} \cdot \frac{3}{3} - 4$

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (1) = - \frac{1}{3} + \frac{2 \cdot 3}{3} - 4$

Write $- 4$ as a fraction with denominator $1$ .

$f (1) = - \frac{1}{3} + \frac{2 \cdot 3}{3} + \frac{- 4}{1}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$f (1) = - \frac{1}{3} + \frac{2 \cdot 3}{3} + \frac{- 4}{1} \cdot \frac{3}{3}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$f (1) = - \frac{1}{3} + \frac{2 \cdot 3}{3} + \frac{- 4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (1) = \frac{- 1 + 2 \cdot 3 - 4 \cdot 3}{3}$

Simplify each term.

Multiply $2$ by $3$ .

$f (1) = \frac{- 1 + 6 - 4 \cdot 3}{3}$

Multiply $- 4$ by $3$ .

$f (1) = \frac{- 1 + 6 - 12}{3}$

Simplify the expression.

Add $- 1$ and $6$ .

$f (1) = \frac{5 - 12}{3}$

Subtract $12$ from $5$ .

$f (1) = \frac{- 7}{3}$

Move the negative in front of the fraction.

$f (1) = - \frac{7}{3}$

The final answer is $- \frac{7}{3}$ .

$- \frac{7}{3}$

The $y$ value at $x = 1$ is $- \frac{7}{3}$ .

$y = - \frac{7}{3}$

Replace the variable $x$ with $6$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (6) = - \frac{36}{3} + 2 (6) - 4$

Divide $36$ by $3$ .

$f (6) = - 1 \cdot 12 + 2 (6) - 4$

Multiply $- 1$ by $12$ .

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

Multiply $2$ by $6$ .

$f (6) = - 12 + 12 - 4$

Simplify by adding and subtracting.

Add $- 12$ and $12$ .

$f (6) = 0 - 4$

Subtract $4$ from $0$ .

$f (6) = - 4$

The final answer is $- 4$ .

$- 4$

The $y$ value at $x = 6$ is $- 4$ .

$y = - 4$

Replace the variable $x$ with $4$ in the expression.

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

Simplify the result.

Simplify each term.

Raise $4$ to the power of $2$ .

$f (4) = - \frac{16}{3} + 2 (4) - 4$

Multiply $2$ by $4$ .

$f (4) = - \frac{16}{3} + 8 - 4$

Find the common denominator.

Write $8$ as a fraction with denominator $1$ .

$f (4) = - \frac{16}{3} + \frac{8}{1} - 4$

Multiply $\frac{8}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{16}{3} + \frac{8}{1} \cdot \frac{3}{3} - 4$

Multiply $\frac{8}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{16}{3} + \frac{8 \cdot 3}{3} - 4$

Write $- 4$ as a fraction with denominator $1$ .

$f (4) = - \frac{16}{3} + \frac{8 \cdot 3}{3} + \frac{- 4}{1}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{16}{3} + \frac{8 \cdot 3}{3} + \frac{- 4}{1} \cdot \frac{3}{3}$

Multiply $\frac{- 4}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{16}{3} + \frac{8 \cdot 3}{3} + \frac{- 4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (4) = \frac{- 16 + 8 \cdot 3 - 4 \cdot 3}{3}$

Simplify each term.

Multiply $8$ by $3$ .

$f (4) = \frac{- 16 + 24 - 4 \cdot 3}{3}$

Multiply $- 4$ by $3$ .

$f (4) = \frac{- 16 + 24 - 12}{3}$

Simplify the expression.

Add $- 16$ and $24$ .

$f (4) = \frac{8 - 12}{3}$

Subtract $12$ from $8$ .

$f (4) = \frac{- 4}{3}$

Move the negative in front of the fraction.

$f (4) = - \frac{4}{3}$

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

The $y$ value at $x = 4$ is $- \frac{4}{3}$ .

$y = - \frac{4}{3}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & - \frac{7}{3} \\ 2 & - \frac{4}{3} \\ 3 & - 1 \\ 4 & - \frac{4}{3} \\ 6 & - 4 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - \frac{7}{4})$

Axis of Symmetry: $x = 3$

Directrix: $y = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 1 & - \frac{7}{3} \\ 2 & - \frac{4}{3} \\ 3 & - 1 \\ 4 & - \frac{4}{3} \\ 6 & - 4 \end{array}$

Step 5

Graph -1/3x^2+2x-4

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