Graph A(x)=-x(x-80)

$A (x) = - x (x - 80)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify $- x (x - 80)$ .

Apply the distributive property.

$y = - x \cdot x - x \cdot - 80$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y = - (x \cdot x) - x \cdot - 80$

Multiply $x$ by $x$ .

$y = - x^{2} - x \cdot - 80$

Multiply $- 80$ by $- 1$ .

$y = - x^{2} + 80 x$

Complete the square for $- x^{2} + 80 x$ .

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

$a = - 1$

$b = 80$

$c = 0$

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

$d = \frac{80}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $80$ and $2$ .

Factor $2$ out of $80$ .

$d = \frac{2 \cdot 40}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{40}{- 1}$ .

$d = - 1 \cdot 40$

Multiply $- 1$ by $40$ .

$d = - 40$

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

$e = 0 - \frac{80^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $80$ to the power of $2$ .

$e = 0 - \frac{6400}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 0 - \frac{6400}{- 4}$

Divide $6400$ by $- 4$ .

$e = 0 - - 1600$

Multiply $- 1$ by $- 1600$ .

$e = 0 + 1600$

Add $0$ and $1600$ .

$e = 1600$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 40)}^{2} + 1600$ .

$- {(x - 40)}^{2} + 1600$

Set $y$ equal to the new right side.

$y = - {(x - 40)}^{2} + 1600$

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

$a = - 1$

$h = 40$

$k = 1600$

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

Opens Down

Find the vertex $(h, k)$ .

$(40, 1600)$

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.

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

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

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

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

$(40, \frac{6399}{4})$

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

$x = 40$

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.

$y = \frac{6401}{4}$

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

Direction: Opens Down

Vertex: $(40, 1600)$

Focus: $(40, \frac{6399}{4})$

Axis of Symmetry: $x = 40$

Directrix: $y = \frac{6401}{4}$

Direction: Opens Down

Vertex: $(40, 1600)$

Focus: $(40, \frac{6399}{4})$

Axis of Symmetry: $x = 40$

Directrix: $y = \frac{6401}{4}$

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 $39$ in the expression.

$f (39) = - {(39)}^{2} + 80 (39)$

Simplify the result.

Simplify each term.

Raise $39$ to the power of $2$ .

$f (39) = - 1 \cdot 1521 + 80 (39)$

Multiply $- 1$ by $1521$ .

$f (39) = - 1521 + 80 (39)$

Multiply $80$ by $39$ .

$f (39) = - 1521 + 3120$

Add $- 1521$ and $3120$ .

$f (39) = 1599$

The final answer is $1599$ .

$1599$

The $y$ value at $x = 39$ is $1599$ .

$y = 1599$

Replace the variable $x$ with $38$ in the expression.

$f (38) = - {(38)}^{2} + 80 (38)$

Simplify the result.

Simplify each term.

Raise $38$ to the power of $2$ .

$f (38) = - 1 \cdot 1444 + 80 (38)$

Multiply $- 1$ by $1444$ .

$f (38) = - 1444 + 80 (38)$

Multiply $80$ by $38$ .

$f (38) = - 1444 + 3040$

Add $- 1444$ and $3040$ .

$f (38) = 1596$

The final answer is $1596$ .

$1596$

The $y$ value at $x = 38$ is $1596$ .

$y = 1596$

Replace the variable $x$ with $41$ in the expression.

$f (41) = - {(41)}^{2} + 80 (41)$

Simplify the result.

Simplify each term.

Raise $41$ to the power of $2$ .

$f (41) = - 1 \cdot 1681 + 80 (41)$

Multiply $- 1$ by $1681$ .

$f (41) = - 1681 + 80 (41)$

Multiply $80$ by $41$ .

$f (41) = - 1681 + 3280$

Add $- 1681$ and $3280$ .

$f (41) = 1599$

The final answer is $1599$ .

$1599$

The $y$ value at $x = 41$ is $1599$ .

$y = 1599$

Replace the variable $x$ with $42$ in the expression.

$f (42) = - {(42)}^{2} + 80 (42)$

Simplify the result.

Simplify each term.

Raise $42$ to the power of $2$ .

$f (42) = - 1 \cdot 1764 + 80 (42)$

Multiply $- 1$ by $1764$ .

$f (42) = - 1764 + 80 (42)$

Multiply $80$ by $42$ .

$f (42) = - 1764 + 3360$

Add $- 1764$ and $3360$ .

$f (42) = 1596$

The final answer is $1596$ .

$1596$

The $y$ value at $x = 42$ is $1596$ .

$y = 1596$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(40, 1600)$

Focus: $(40, \frac{6399}{4})$

Axis of Symmetry: $x = 40$

Directrix: $y = \frac{6401}{4}$

$\begin{array}{c} x & y \\ 38 & 1596 \\ 39 & 1599 \\ 40 & 1600 \\ 41 & 1599 \\ 42 & 1596 \end{array}$

Step 4

Graph A(x)=-x(x-80)

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