Graph f(x)=-1/3x^2+4/3x-16/3

$f (x) = - \frac{1}{3} x^{2} + \frac{4}{3} x - \frac{16}{3}$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify each term.

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

$y = - \frac{x^{2}}{3} + \frac{4}{3} x - \frac{16}{3}$

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

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{16}{3}$

Complete the square for $- \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{16}{3}$ .

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

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

$b = \frac{4}{3}$

$c = - \frac{16}{3}$

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply $- 1$ by $3$ .

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

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

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

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

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

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

$e = - \frac{16}{3} - \frac{{(\frac{4}{3})}^{2}}{4 (- \frac{1}{3})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{4}{3}$ .

$e = - \frac{16}{3} - \frac{\frac{4^{2}}{3^{2}}}{4 (- \frac{1}{3})}$

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

$e = - \frac{16}{3} - \frac{\frac{16}{3^{2}}}{4 (- \frac{1}{3})}$

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

$e = - \frac{16}{3} - \frac{\frac{16}{9}}{4 (- \frac{1}{3})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{16}{3} - \frac{\frac{16}{9}}{- 4 (\frac{1}{3})}$

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

$e = - \frac{16}{3} - \frac{\frac{16}{9}}{\frac{- 4}{3}}$

Move the negative in front of the fraction.

$e = - \frac{16}{3} - \frac{\frac{16}{9}}{- \frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{16}{3} - (\frac{16}{9} (- \frac{3}{4}))$

Cancel the common factor of $4$ .

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

$e = - \frac{16}{3} - (\frac{16}{9} \cdot \frac{- 3}{4})$

Factor $4$ out of $16$ .

$e = - \frac{16}{3} - (\frac{4 (4)}{9} \cdot \frac{- 3}{4})$

Cancel the common factor.

$e = - \frac{16}{3} - (\frac{4 \cdot 4}{9} \cdot \frac{- 3}{4})$

Rewrite the expression.

$e = - \frac{16}{3} - (\frac{4}{9} \cdot - 3)$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = - \frac{16}{3} - (\frac{4}{3 (3)} \cdot - 3)$

Factor $3$ out of $- 3$ .

$e = - \frac{16}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot - 1))$

Cancel the common factor.

$e = - \frac{16}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot - 1))$

Rewrite the expression.

$e = - \frac{16}{3} - (\frac{4}{3} \cdot - 1)$

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

$e = - \frac{16}{3} - \frac{4 \cdot - 1}{3}$

Multiply $4$ by $- 1$ .

$e = - \frac{16}{3} - \frac{- 4}{3}$

Move the negative in front of the fraction.

$e = - \frac{16}{3} - - \frac{4}{3}$

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

Multiply $- 1$ by $- 1$ .

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

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

$e = - \frac{16}{3} + \frac{4}{3}$

Combine the numerators over the common denominator.

$e = \frac{- 16 + 4}{3}$

Add $- 16$ and $4$ .

$e = \frac{- 12}{3}$

Divide $- 12$ by $3$ .

$e = - 4$

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

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

Set $y$ equal to the new right side.

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

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

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

$h = 2$

$k = - 4$

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

Opens Down

Find the vertex $(h, k)$ .

$(2, - 4)$

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.

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

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.

$(2, - \frac{19}{4})$

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

$x = 2$

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.

$y = - \frac{13}{4}$

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

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{19}{4})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{13}{4}$

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{19}{4})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{13}{4}$

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.

$f (1) = - \frac{{(1)}^{2}}{3} + \frac{4 (1)}{3} - \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (1) = \frac{- {(1)}^{2} + 4 (1) - 16}{3}$

Simplify each term.

One to any power is one.

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

Multiply $- 1$ by $1$ .

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

Multiply $4$ by $1$ .

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

Simplify the expression.

Add $- 1$ and $4$ .

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

Subtract $16$ from $3$ .

$f (1) = \frac{- 13}{3}$

Move the negative in front of the fraction.

$f (1) = - \frac{13}{3}$

The final answer is $- \frac{13}{3}$ .

$- \frac{13}{3}$

The $y$ value at $x = 1$ is $- \frac{13}{3}$ .

$y = - \frac{13}{3}$

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{{(0)}^{2}}{3} + \frac{4 (0)}{3} - \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (0) = \frac{- {(0)}^{2} + 4 (0) - 16}{3}$

Simplify each term.

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

$f (0) = \frac{- 0 + 4 (0) - 16}{3}$

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 + 4 (0) - 16}{3}$

Multiply $4$ by $0$ .

$f (0) = \frac{0 + 0 - 16}{3}$

Simplify the expression.

Add $0$ and $0$ .

$f (0) = \frac{0 - 16}{3}$

Subtract $16$ from $0$ .

$f (0) = \frac{- 16}{3}$

Move the negative in front of the fraction.

$f (0) = - \frac{16}{3}$

The final answer is $- \frac{16}{3}$ .

$- \frac{16}{3}$

The $y$ value at $x = 0$ is $- \frac{16}{3}$ .

$y = - \frac{16}{3}$

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

$f (5) = - \frac{{(5)}^{2}}{3} + \frac{4 (5)}{3} - \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (5) = \frac{- {(5)}^{2} + 4 (5) - 16}{3}$

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = \frac{- 1 \cdot 25 + 4 (5) - 16}{3}$

Multiply $- 1$ by $25$ .

$f (5) = \frac{- 25 + 4 (5) - 16}{3}$

Multiply $4$ by $5$ .

$f (5) = \frac{- 25 + 20 - 16}{3}$

Simplify the expression.

Add $- 25$ and $20$ .

$f (5) = \frac{- 5 - 16}{3}$

Subtract $16$ from $- 5$ .

$f (5) = \frac{- 21}{3}$

Divide $- 21$ by $3$ .

$f (5) = - 7$

The final answer is $- 7$ .

$- 7$

The $y$ value at $x = 5$ is $- 7$ .

$y = - 7$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (3) = \frac{- 1 \cdot 9 + 4 (3) - 16}{3}$

Multiply $- 1$ by $9$ .

$f (3) = \frac{- 9 + 4 (3) - 16}{3}$

Multiply $4$ by $3$ .

$f (3) = \frac{- 9 + 12 - 16}{3}$

Simplify the expression.

Add $- 9$ and $12$ .

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

Subtract $16$ from $3$ .

$f (3) = \frac{- 13}{3}$

Move the negative in front of the fraction.

$f (3) = - \frac{13}{3}$

The final answer is $- \frac{13}{3}$ .

$- \frac{13}{3}$

The $y$ value at $x = 3$ is $- \frac{13}{3}$ .

$y = - \frac{13}{3}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - \frac{16}{3} \\ 1 & - \frac{13}{3} \\ 2 & - 4 \\ 3 & - \frac{13}{3} \\ 5 & - 7 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{19}{4})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{13}{4}$

$\begin{array}{c} x & y \\ 0 & - \frac{16}{3} \\ 1 & - \frac{13}{3} \\ 2 & - 4 \\ 3 & - \frac{13}{3} \\ 5 & - 7 \end{array}$

Step 4

Graph f(x)=-1/3x^2+4/3x-16/3

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