Graph g(x)=1/3(x-6)^2+1

$g (x) = \frac{1}{3} {(x - 6)}^{2} + 1$

Find the properties of the given parabola.

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

Combine $\frac{1}{3}$ and ${(x - 6)}^{2}$ .

$y = \frac{{(x - 6)}^{2}}{3} + 1$

Reorder terms.

$y = \frac{1}{3} \cdot {(x - 6)}^{2} + 1$

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

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

$h = 6$

$k = 1$

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

Opens Up

Find the vertex $(h, k)$ .

$(6, 1)$

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.

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.

$(6, \frac{7}{4})$

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

$x = 6$

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 Up

Vertex: $(6, 1)$

Focus: $(6, \frac{7}{4})$

Axis of Symmetry: $x = 6$

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

Direction: Opens Up

Vertex: $(6, 1)$

Focus: $(6, \frac{7}{4})$

Axis of Symmetry: $x = 6$

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

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 $5$ in the expression.

$f (5) = \frac{{(5)}^{2}}{3} - 4 \cdot 5 + 13$

Simplify the result.

Simplify each term.

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

$f (5) = \frac{25}{3} - 4 \cdot 5 + 13$

Multiply $- 4$ by $5$ .

$f (5) = \frac{25}{3} - 20 + 13$

Find the common denominator.

Write $- 20$ as a fraction with denominator $1$ .

$f (5) = \frac{25}{3} + \frac{- 20}{1} + 13$

Multiply $\frac{- 20}{1}$ by $\frac{3}{3}$ .

$f (5) = \frac{25}{3} + \frac{- 20}{1} \cdot \frac{3}{3} + 13$

Multiply $\frac{- 20}{1}$ by $\frac{3}{3}$ .

$f (5) = \frac{25}{3} + \frac{- 20 \cdot 3}{3} + 13$

Write $13$ as a fraction with denominator $1$ .

$f (5) = \frac{25}{3} + \frac{- 20 \cdot 3}{3} + \frac{13}{1}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (5) = \frac{25}{3} + \frac{- 20 \cdot 3}{3} + \frac{13}{1} \cdot \frac{3}{3}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (5) = \frac{25}{3} + \frac{- 20 \cdot 3}{3} + \frac{13 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (5) = \frac{25 - 20 \cdot 3 + 13 \cdot 3}{3}$

Simplify each term.

Multiply $- 20$ by $3$ .

$f (5) = \frac{25 - 60 + 13 \cdot 3}{3}$

Multiply $13$ by $3$ .

$f (5) = \frac{25 - 60 + 39}{3}$

Simplify by adding and subtracting.

Subtract $60$ from $25$ .

$f (5) = \frac{- 35 + 39}{3}$

Add $- 35$ and $39$ .

$f (5) = \frac{4}{3}$

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

$\frac{4}{3}$

The $y$ value at $x = 5$ is $\frac{4}{3}$ .

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

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

$f (4) = \frac{{(4)}^{2}}{3} - 4 \cdot 4 + 13$

Simplify the result.

Simplify each term.

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

$f (4) = \frac{16}{3} - 4 \cdot 4 + 13$

Multiply $- 4$ by $4$ .

$f (4) = \frac{16}{3} - 16 + 13$

Find the common denominator.

Write $- 16$ as a fraction with denominator $1$ .

$f (4) = \frac{16}{3} + \frac{- 16}{1} + 13$

Multiply $\frac{- 16}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{16}{3} + \frac{- 16}{1} \cdot \frac{3}{3} + 13$

Multiply $\frac{- 16}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{16}{3} + \frac{- 16 \cdot 3}{3} + 13$

Write $13$ as a fraction with denominator $1$ .

$f (4) = \frac{16}{3} + \frac{- 16 \cdot 3}{3} + \frac{13}{1}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{16}{3} + \frac{- 16 \cdot 3}{3} + \frac{13}{1} \cdot \frac{3}{3}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (4) = \frac{16}{3} + \frac{- 16 \cdot 3}{3} + \frac{13 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (4) = \frac{16 - 16 \cdot 3 + 13 \cdot 3}{3}$

Simplify each term.

Multiply $- 16$ by $3$ .

$f (4) = \frac{16 - 48 + 13 \cdot 3}{3}$

Multiply $13$ by $3$ .

$f (4) = \frac{16 - 48 + 39}{3}$

Simplify by adding and subtracting.

Subtract $48$ from $16$ .

$f (4) = \frac{- 32 + 39}{3}$

Add $- 32$ and $39$ .

$f (4) = \frac{7}{3}$

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

$\frac{7}{3}$

The $y$ value at $x = 4$ is $\frac{7}{3}$ .

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

Replace the variable $x$ with $9$ in the expression.

$f (9) = \frac{{(9)}^{2}}{3} - 4 \cdot 9 + 13$

Simplify the result.

Simplify each term.

Raise $9$ to the power of $2$ .

$f (9) = \frac{81}{3} - 4 \cdot 9 + 13$

Divide $81$ by $3$ .

$f (9) = 27 - 4 \cdot 9 + 13$

Multiply $- 4$ by $9$ .

$f (9) = 27 - 36 + 13$

Simplify by adding and subtracting.

Subtract $36$ from $27$ .

$f (9) = - 9 + 13$

Add $- 9$ and $13$ .

$f (9) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 9$ is $4$ .

$y = 4$

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{{(7)}^{2}}{3} - 4 \cdot 7 + 13$

Simplify the result.

Simplify each term.

Raise $7$ to the power of $2$ .

$f (7) = \frac{49}{3} - 4 \cdot 7 + 13$

Multiply $- 4$ by $7$ .

$f (7) = \frac{49}{3} - 28 + 13$

Find the common denominator.

Write $- 28$ as a fraction with denominator $1$ .

$f (7) = \frac{49}{3} + \frac{- 28}{1} + 13$

Multiply $\frac{- 28}{1}$ by $\frac{3}{3}$ .

$f (7) = \frac{49}{3} + \frac{- 28}{1} \cdot \frac{3}{3} + 13$

Multiply $\frac{- 28}{1}$ by $\frac{3}{3}$ .

$f (7) = \frac{49}{3} + \frac{- 28 \cdot 3}{3} + 13$

Write $13$ as a fraction with denominator $1$ .

$f (7) = \frac{49}{3} + \frac{- 28 \cdot 3}{3} + \frac{13}{1}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (7) = \frac{49}{3} + \frac{- 28 \cdot 3}{3} + \frac{13}{1} \cdot \frac{3}{3}$

Multiply $\frac{13}{1}$ by $\frac{3}{3}$ .

$f (7) = \frac{49}{3} + \frac{- 28 \cdot 3}{3} + \frac{13 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (7) = \frac{49 - 28 \cdot 3 + 13 \cdot 3}{3}$

Simplify each term.

Multiply $- 28$ by $3$ .

$f (7) = \frac{49 - 84 + 13 \cdot 3}{3}$

Multiply $13$ by $3$ .

$f (7) = \frac{49 - 84 + 39}{3}$

Simplify by adding and subtracting.

Subtract $84$ from $49$ .

$f (7) = \frac{- 35 + 39}{3}$

Add $- 35$ and $39$ .

$f (7) = \frac{4}{3}$

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

$\frac{4}{3}$

The $y$ value at $x = 7$ is $\frac{4}{3}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & \frac{7}{3} \\ 5 & \frac{4}{3} \\ 6 & 1 \\ 7 & \frac{4}{3} \\ 9 & 4 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(6, 1)$

Focus: $(6, \frac{7}{4})$

Axis of Symmetry: $x = 6$

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

$\begin{array}{c} x & y \\ 4 & \frac{7}{3} \\ 5 & \frac{4}{3} \\ 6 & 1 \\ 7 & \frac{4}{3} \\ 9 & 4 \end{array}$

Graph g(x)=1/3(x-6)^2+1

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