Graph g(x)=1/20x(x-100)

$g (x) = \frac{1}{20} x (x - 100)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

Remove parentheses.

$y = \frac{1}{20} \cdot (x (x - 100))$

Remove parentheses.

$y = \frac{1}{20} \cdot (x (x - 100))$

Simplify $\frac{1}{20} \cdot (x (x - 100))$ .

Apply the distributive property.

$y = \frac{1}{20} \cdot (x \cdot x + x \cdot - 100)$

Simplify the expression.

Multiply $x$ by $x$ .

$y = \frac{1}{20} \cdot (x^{2} + x \cdot - 100)$

Move $- 100$ to the left of $x$ .

$y = \frac{1}{20} \cdot (x^{2} - 100 \cdot x)$

Apply the distributive property.

$y = \frac{1}{20} x^{2} + \frac{1}{20} (- 100 x)$

Combine $\frac{1}{20}$ and $x^{2}$ .

$y = \frac{x^{2}}{20} + \frac{1}{20} (- 100 x)$

Cancel the common factor of $20$ .

Factor $20$ out of $- 100 x$ .

$y = \frac{x^{2}}{20} + \frac{1}{20} (20 (- 5 x))$

Cancel the common factor.

$y = \frac{x^{2}}{20} + \frac{1}{20} (20 (- 5 x))$

Rewrite the expression.

$y = \frac{x^{2}}{20} - 5 x$

Complete the square for $\frac{x^{2}}{20} - 5 x$ .

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

$a = \frac{1}{20}$

$b = - 5$

$c = 0$

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

$d = \frac{- 5}{2 (\frac{1}{20})}$

Simplify the right side.

Combine $2$ and $\frac{1}{20}$ .

$d = \frac{- 5}{\frac{2}{20}}$

Cancel the common factor of $2$ and $20$ .

Factor $2$ out of $2$ .

$d = \frac{- 5}{\frac{2 (1)}{20}}$

Cancel the common factors.

Factor $2$ out of $20$ .

$d = \frac{- 5}{\frac{2 \cdot 1}{2 \cdot 10}}$

Cancel the common factor.

$d = \frac{- 5}{\frac{2 \cdot 1}{2 \cdot 10}}$

Rewrite the expression.

$d = \frac{- 5}{\frac{1}{10}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - 5 \cdot 10$

Multiply $- 5$ by $10$ .

$d = - 50$

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

$e = 0 - \frac{{(- 5)}^{2}}{4 (\frac{1}{20})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{25}{4 (\frac{1}{20})}$

Combine $4$ and $\frac{1}{20}$ .

$e = 0 - \frac{25}{\frac{4}{20}}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$e = 0 - \frac{25}{\frac{4 (1)}{20}}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = 0 - \frac{25}{\frac{4 \cdot 1}{4 \cdot 5}}$

Cancel the common factor.

$e = 0 - \frac{25}{\frac{4 \cdot 1}{4 \cdot 5}}$

Rewrite the expression.

$e = 0 - \frac{25}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (25 \cdot 5)$

Multiply $- (25 \cdot 5)$ .

Multiply $25$ by $5$ .

$e = 0 - 1 \cdot 125$

Multiply $- 1$ by $125$ .

$e = 0 - 125$

Subtract $125$ from $0$ .

$e = - 125$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{20} {(x - 50)}^{2} - 125$ .

$\frac{1}{20} {(x - 50)}^{2} - 125$

Set $y$ equal to the new right side.

$y = \frac{1}{20} \cdot {(x - 50)}^{2} - 125$

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

$a = \frac{1}{20}$

$h = 50$

$k = - 125$

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

Opens Up

Find the vertex $(h, k)$ .

$(50, - 125)$

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.

$\frac{1}{4 \cdot \frac{1}{20}}$

Simplify.

Combine $4$ and $\frac{1}{20}$ .

$\frac{1}{\frac{4}{20}}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{20}}$

Cancel the common factors.

Factor $4$ out of $20$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Rewrite the expression.

$\frac{1}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 5$

Multiply $5$ by $1$ .

$5$

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.

$(50, - 120)$

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

$x = 50$

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.

$y = - 130$

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

Direction: Opens Up

Vertex: $(50, - 125)$

Focus: $(50, - 120)$

Axis of Symmetry: $x = 50$

Directrix: $y = - 130$

Direction: Opens Up

Vertex: $(50, - 125)$

Focus: $(50, - 120)$

Axis of Symmetry: $x = 50$

Directrix: $y = - 130$

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 $49$ in the expression.

$f (49) = \frac{{(49)}^{2}}{20} - 5 \cdot 49$

Simplify the result.

Simplify each term.

Raise $49$ to the power of $2$ .

$f (49) = \frac{2401}{20} - 5 \cdot 49$

Multiply $- 5$ by $49$ .

$f (49) = \frac{2401}{20} - 245$

To write $- 245$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$f (49) = \frac{2401}{20} - 245 \cdot \frac{20}{20}$

Combine $- 245$ and $\frac{20}{20}$ .

$f (49) = \frac{2401}{20} + \frac{- 245 \cdot 20}{20}$

Combine the numerators over the common denominator.

$f (49) = \frac{2401 - 245 \cdot 20}{20}$

Simplify the numerator.

Multiply $- 245$ by $20$ .

$f (49) = \frac{2401 - 4900}{20}$

Subtract $4900$ from $2401$ .

$f (49) = \frac{- 2499}{20}$

Move the negative in front of the fraction.

$f (49) = - \frac{2499}{20}$

The final answer is $- \frac{2499}{20}$ .

$- \frac{2499}{20}$

The $y$ value at $x = 49$ is $- \frac{2499}{20}$ .

$y = - \frac{2499}{20}$

Replace the variable $x$ with $48$ in the expression.

$f (48) = \frac{{(48)}^{2}}{20} - 5 \cdot 48$

Simplify the result.

Simplify each term.

Raise $48$ to the power of $2$ .

$f (48) = \frac{2304}{20} - 5 \cdot 48$

Cancel the common factor of $2304$ and $20$ .

Factor $4$ out of $2304$ .

$f (48) = \frac{4 (576)}{20} - 5 \cdot 48$

Cancel the common factors.

Factor $4$ out of $20$ .

$f (48) = \frac{4 \cdot 576}{4 \cdot 5} - 5 \cdot 48$

Cancel the common factor.

$f (48) = \frac{4 \cdot 576}{4 \cdot 5} - 5 \cdot 48$

Rewrite the expression.

$f (48) = \frac{576}{5} - 5 \cdot 48$

Multiply $- 5$ by $48$ .

$f (48) = \frac{576}{5} - 240$

To write $- 240$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (48) = \frac{576}{5} - 240 \cdot \frac{5}{5}$

Combine $- 240$ and $\frac{5}{5}$ .

$f (48) = \frac{576}{5} + \frac{- 240 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (48) = \frac{576 - 240 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 240$ by $5$ .

$f (48) = \frac{576 - 1200}{5}$

Subtract $1200$ from $576$ .

$f (48) = \frac{- 624}{5}$

Move the negative in front of the fraction.

$f (48) = - \frac{624}{5}$

The final answer is $- \frac{624}{5}$ .

$- \frac{624}{5}$

The $y$ value at $x = 48$ is $- \frac{624}{5}$ .

$y = - \frac{624}{5}$

Replace the variable $x$ with $51$ in the expression.

$f (51) = \frac{{(51)}^{2}}{20} - 5 \cdot 51$

Simplify the result.

Simplify each term.

Raise $51$ to the power of $2$ .

$f (51) = \frac{2601}{20} - 5 \cdot 51$

Multiply $- 5$ by $51$ .

$f (51) = \frac{2601}{20} - 255$

To write $- 255$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$f (51) = \frac{2601}{20} - 255 \cdot \frac{20}{20}$

Combine $- 255$ and $\frac{20}{20}$ .

$f (51) = \frac{2601}{20} + \frac{- 255 \cdot 20}{20}$

Combine the numerators over the common denominator.

$f (51) = \frac{2601 - 255 \cdot 20}{20}$

Simplify the numerator.

Multiply $- 255$ by $20$ .

$f (51) = \frac{2601 - 5100}{20}$

Subtract $5100$ from $2601$ .

$f (51) = \frac{- 2499}{20}$

Move the negative in front of the fraction.

$f (51) = - \frac{2499}{20}$

The final answer is $- \frac{2499}{20}$ .

$- \frac{2499}{20}$

The $y$ value at $x = 51$ is $- \frac{2499}{20}$ .

$y = - \frac{2499}{20}$

Replace the variable $x$ with $52$ in the expression.

$f (52) = \frac{{(52)}^{2}}{20} - 5 \cdot 52$

Simplify the result.

Simplify each term.

Raise $52$ to the power of $2$ .

$f (52) = \frac{2704}{20} - 5 \cdot 52$

Cancel the common factor of $2704$ and $20$ .

Factor $4$ out of $2704$ .

$f (52) = \frac{4 (676)}{20} - 5 \cdot 52$

Cancel the common factors.

Factor $4$ out of $20$ .

$f (52) = \frac{4 \cdot 676}{4 \cdot 5} - 5 \cdot 52$

Cancel the common factor.

$f (52) = \frac{4 \cdot 676}{4 \cdot 5} - 5 \cdot 52$

Rewrite the expression.

$f (52) = \frac{676}{5} - 5 \cdot 52$

Multiply $- 5$ by $52$ .

$f (52) = \frac{676}{5} - 260$

To write $- 260$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (52) = \frac{676}{5} - 260 \cdot \frac{5}{5}$

Combine $- 260$ and $\frac{5}{5}$ .

$f (52) = \frac{676}{5} + \frac{- 260 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (52) = \frac{676 - 260 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 260$ by $5$ .

$f (52) = \frac{676 - 1300}{5}$

Subtract $1300$ from $676$ .

$f (52) = \frac{- 624}{5}$

Move the negative in front of the fraction.

$f (52) = - \frac{624}{5}$

The final answer is $- \frac{624}{5}$ .

$- \frac{624}{5}$

The $y$ value at $x = 52$ is $- \frac{624}{5}$ .

$y = - \frac{624}{5}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 48 & - \frac{624}{5} \\ 49 & - \frac{2499}{20} \\ 50 & - 125 \\ 51 & - \frac{2499}{20} \\ 52 & - \frac{624}{5} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(50, - 125)$

Focus: $(50, - 120)$

Axis of Symmetry: $x = 50$

Directrix: $y = - 130$

$\begin{array}{c} x & y \\ 48 & - \frac{624}{5} \\ 49 & - \frac{2499}{20} \\ 50 & - 125 \\ 51 & - \frac{2499}{20} \\ 52 & - \frac{624}{5} \end{array}$

Step 4

Graph g(x)=1/20x(x-100)

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