Graph y=1/3(x-4)^2

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

Step 1

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

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

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

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $- 4$ by $- 4$ .

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

Subtract $4 x$ from $- 4 x$ .

$y = \frac{1}{3} \cdot (x^{2} - 8 x + 16)$

Apply the distributive property.

$y = \frac{1}{3} x^{2} + \frac{1}{3} (- 8 x) + \frac{1}{3} \cdot 16$

Simplify.

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

$y = \frac{x^{2}}{3} + \frac{1}{3} (- 8 x) + \frac{1}{3} \cdot 16$

Multiply $\frac{1}{3} (- 8 x)$ .

Combine $- 8$ and $\frac{1}{3}$ .

$y = \frac{x^{2}}{3} + \frac{- 8}{3} x + \frac{1}{3} \cdot 16$

Combine $\frac{- 8}{3}$ and $x$ .

$y = \frac{x^{2}}{3} + \frac{- 8 x}{3} + \frac{1}{3} \cdot 16$

Combine $\frac{1}{3}$ and $16$ .

$y = \frac{x^{2}}{3} + \frac{- 8 x}{3} + \frac{16}{3}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{3} - \frac{8 x}{3} + \frac{16}{3}$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{x^{2}}{3} - \frac{8 x}{3} + \frac{16}{3}$ .

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

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

$b = - \frac{8}{3}$

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

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

$d = \frac{- \frac{8}{3}}{2 (\frac{1}{3})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{8}{3} \cdot \frac{1}{2 (\frac{1}{3})}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$d = \frac{- 8}{3} \cdot \frac{1}{2 (\frac{1}{3})}$

Factor $2$ out of $- 8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = \frac{16}{3} - \frac{{(- \frac{8}{3})}^{2}}{4 (\frac{1}{3})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{8}{3}$ .

$e = \frac{16}{3} - \frac{{(- 1)}^{2} {(\frac{8}{3})}^{2}}{4 (\frac{1}{3})}$

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

$e = \frac{16}{3} - \frac{1 {(\frac{8}{3})}^{2}}{4 (\frac{1}{3})}$

Apply the product rule to $\frac{8}{3}$ .

$e = \frac{16}{3} - \frac{1 \frac{8^{2}}{3^{2}}}{4 (\frac{1}{3})}$

Raise $8$ to the power of $2$ .

$e = \frac{16}{3} - \frac{1 \frac{64}{3^{2}}}{4 (\frac{1}{3})}$

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

$e = \frac{16}{3} - \frac{1 (\frac{64}{9})}{4 (\frac{1}{3})}$

Multiply $\frac{64}{9}$ by $1$ .

$e = \frac{16}{3} - \frac{\frac{64}{9}}{4 (\frac{1}{3})}$

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

$e = \frac{16}{3} - \frac{\frac{64}{9}}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{16}{3} - (\frac{64}{9} \cdot \frac{3}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Subtract $16$ from $16$ .

$e = \frac{0}{3}$

Divide $0$ by $3$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{3} {(x - 4)}^{2} + 0$ .

$\frac{1}{3} {(x - 4)}^{2} + 0$

Set $y$ equal to the new right side.

$y = \frac{1}{3} \cdot {(x - 4)}^{2} + 0$

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

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

$h = 4$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(4, 0)$

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.

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

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

$x = 4$

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.

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

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

Direction: Opens Up

Vertex: $(4, 0)$

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

Axis of Symmetry: $x = 4$

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

Direction: Opens Up

Vertex: $(4, 0)$

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

Axis of Symmetry: $x = 4$

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

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 $3$ in the expression.

$f (3) = \frac{{(3)}^{2}}{3} - \frac{8 (3)}{3} + \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (3) = \frac{{(3)}^{2} - 8 \cdot 3 + 16}{3}$

Simplify each term.

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

$f (3) = \frac{9 - 8 \cdot 3 + 16}{3}$

Multiply $- 8$ by $3$ .

$f (3) = \frac{9 - 24 + 16}{3}$

Simplify by adding and subtracting.

Subtract $24$ from $9$ .

$f (3) = \frac{- 15 + 16}{3}$

Add $- 15$ and $16$ .

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

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

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

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

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

$f (2) = \frac{{(2)}^{2}}{3} - \frac{8 (2)}{3} + \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (2) = \frac{{(2)}^{2} - 8 \cdot 2 + 16}{3}$

Simplify each term.

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

$f (2) = \frac{4 - 8 \cdot 2 + 16}{3}$

Multiply $- 8$ by $2$ .

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

Simplify by adding and subtracting.

Subtract $16$ from $4$ .

$f (2) = \frac{- 12 + 16}{3}$

Add $- 12$ and $16$ .

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

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 $7$ in the expression.

$f (7) = \frac{{(7)}^{2}}{3} - \frac{8 (7)}{3} + \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (7) = \frac{{(7)}^{2} - 8 \cdot 7 + 16}{3}$

Simplify each term.

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

$f (7) = \frac{49 - 8 \cdot 7 + 16}{3}$

Multiply $- 8$ by $7$ .

$f (7) = \frac{49 - 56 + 16}{3}$

Simplify the expression.

Subtract $56$ from $49$ .

$f (7) = \frac{- 7 + 16}{3}$

Add $- 7$ and $16$ .

$f (7) = \frac{9}{3}$

Divide $9$ by $3$ .

$f (7) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 7$ is $3$ .

$y = 3$

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

$f (5) = \frac{{(5)}^{2}}{3} - \frac{8 (5)}{3} + \frac{16}{3}$

Simplify the result.

Combine the numerators over the common denominator.

$f (5) = \frac{{(5)}^{2} - 8 \cdot 5 + 16}{3}$

Simplify each term.

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

$f (5) = \frac{25 - 8 \cdot 5 + 16}{3}$

Multiply $- 8$ by $5$ .

$f (5) = \frac{25 - 40 + 16}{3}$

Simplify by adding and subtracting.

Subtract $40$ from $25$ .

$f (5) = \frac{- 15 + 16}{3}$

Add $- 15$ and $16$ .

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

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

The $y$ value at $x = 5$ is $\frac{1}{3}$ .

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

Graph the parabola using its properties and the selected points.

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

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, 0)$

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

Axis of Symmetry: $x = 4$

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

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

Step 5

Graph y=1/3(x-4)^2

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