Graph f(x)=0.04(8x-x^2)

$f (x) = 0.04 (8 x - x^{2})$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify $0.04 (8 x - x^{2})$ .

Apply the distributive property.

$y = 0.04 (8 x) + 0.04 (- x^{2})$

Simplify the expression.

Multiply $8$ by $0.04$ .

$y = 0.32 x + 0.04 (- x^{2})$

Multiply $- 1$ by $0.04$ .

$y = 0.32 x - 0.04 x^{2}$

Reorder $0.32 x$ and $- 0.04 x^{2}$ .

$y = - 0.04 x^{2} + 0.32 x$

Complete the square for $- 0.04 x^{2} + 0.32 x$ .

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

$a = - 0.04$

$b = 0.32$

$c = 0$

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

$d = \frac{0.32}{2 \cdot - 0.04}$

Simplify the right side.

Multiply $2$ by $- 0.04$ .

$d = \frac{0.32}{- 0.08}$

Divide $0.32$ by $- 0.08$ .

$d = - 4$

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

$e = 0 - \frac{{0.32}^{2}}{4 \cdot - 0.04}$

Simplify the right side.

Simplify each term.

Raise $0.32$ to the power of $2$ .

$e = 0 - \frac{0.1024}{4 \cdot - 0.04}$

Multiply $4$ by $- 0.04$ .

$e = 0 - \frac{0.1024}{- 0.16}$

Divide $0.1024$ by $- 0.16$ .

$e = 0 - - 0.64$

Multiply $- 1$ by $- 0.64$ .

$e = 0 + 0.64$

Add $0$ and $0.64$ .

$e = 0.64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 0.04 {(x - 4)}^{2} + 0.64$ .

$- 0.04 {(x - 4)}^{2} + 0.64$

Set $y$ equal to the new right side.

$y = - 0.04 {(x - 4)}^{2} + 0.64$

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

$a = - 0.04$

$h = 4$

$k = 0.64$

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

Opens Down

Find the vertex $(h, k)$ .

$(4, 0.64)$

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.

$\frac{1}{4 \cdot - 0.04}$

Simplify.

Multiply $4$ by $- 0.04$ .

$\frac{1}{- 0.16}$

Divide $1$ by $- 0.16$ .

$- 6.25$

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.

$(4, - 5.61)$

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.

$y = 6.89$

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

Direction: Opens Down

Vertex: $(4, 0.64)$

Focus: $(4, - 5.61)$

Axis of Symmetry: $x = 4$

Directrix: $y = 6.89$

Direction: Opens Down

Vertex: $(4, 0.64)$

Focus: $(4, - 5.61)$

Axis of Symmetry: $x = 4$

Directrix: $y = 6.89$

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.

$f (3) = - 0.04 {(3)}^{2} + 0.32 (3)$

Simplify the result.

Simplify each term.

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

$f (3) = - 0.04 \cdot 9 + 0.32 (3)$

Multiply $- 0.04$ by $9$ .

$f (3) = - 0.36 + 0.32 (3)$

Multiply $0.32$ by $3$ .

$f (3) = - 0.36 + 0.96$

Add $- 0.36$ and $0.96$ .

$f (3) = 0.6$

The final answer is $0.6$ .

$0.6$

The $y$ value at $x = 3$ is $0.6$ .

$y = 0.6$

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

$f (2) = - 0.04 {(2)}^{2} + 0.32 (2)$

Simplify the result.

Simplify each term.

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

$f (2) = - 0.04 \cdot 4 + 0.32 (2)$

Multiply $- 0.04$ by $4$ .

$f (2) = - 0.16 + 0.32 (2)$

Multiply $0.32$ by $2$ .

$f (2) = - 0.16 + 0.64$

Add $- 0.16$ and $0.64$ .

$f (2) = 0.48$

The final answer is $0.48$ .

$0.48$

The $y$ value at $x = 2$ is $0.48$ .

$y = 0.48$

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

$f (5) = - 0.04 {(5)}^{2} + 0.32 (5)$

Simplify the result.

Simplify each term.

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

$f (5) = - 0.04 \cdot 25 + 0.32 (5)$

Multiply $- 0.04$ by $25$ .

$f (5) = - 1 + 0.32 (5)$

Multiply $0.32$ by $5$ .

$f (5) = - 1 + 1.6$

Add $- 1$ and $1.6$ .

$f (5) = 0.6$

The final answer is $0.6$ .

$0.6$

The $y$ value at $x = 5$ is $0.6$ .

$y = 0.6$

Replace the variable $x$ with $6$ in the expression.

$f (6) = - 0.04 {(6)}^{2} + 0.32 (6)$

Simplify the result.

Simplify each term.

Raise $6$ to the power of $2$ .

$f (6) = - 0.04 \cdot 36 + 0.32 (6)$

Multiply $- 0.04$ by $36$ .

$f (6) = - 1.44 + 0.32 (6)$

Multiply $0.32$ by $6$ .

$f (6) = - 1.44 + 1.92$

Add $- 1.44$ and $1.92$ .

$f (6) = 0.48$

The final answer is $0.48$ .

$0.48$

The $y$ value at $x = 6$ is $0.48$ .

$y = 0.48$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 0.48 \\ 3 & 0.6 \\ 4 & 0.64 \\ 5 & 0.6 \\ 6 & 0.48 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(4, 0.64)$

Focus: $(4, - 5.61)$

Axis of Symmetry: $x = 4$

Directrix: $y = 6.89$

$\begin{array}{c} x & y \\ 2 & 0.48 \\ 3 & 0.6 \\ 4 & 0.64 \\ 5 & 0.6 \\ 6 & 0.48 \end{array}$

Graph f(x)=0.04(8x-x^2)

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