Graph h(x)=-1/5x^2+2x

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

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{5}$ .

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

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

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

$a = - \frac{1}{5}$

$b = 2$

$c = 0$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 5)$

Multiply $5$ by $1$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

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

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

Simplify the right side.

Simplify each term.

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

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

Combine $- 4$ and $\frac{1}{5}$ .

$e = 0 - \frac{4}{\frac{- 4}{5}}$

Move the negative in front of the fraction.

$e = 0 - \frac{4}{- \frac{4}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (4 (- \frac{5}{4}))$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = 0 - (4 (\frac{- 5}{4}))$

Cancel the common factor.

$e = 0 - (4 (\frac{- 5}{4}))$

Rewrite the expression.

$e = 0 - - 5$

Multiply $- 1$ by $- 5$ .

$e = 0 + 5$

Add $0$ and $5$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{5} {(x - 5)}^{2} + 5$ .

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

Set $y$ equal to the new right side.

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

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

$a = - \frac{1}{5}$

$h = 5$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(5, 5)$

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.

$\frac{1}{4 \cdot (- \frac{1}{5})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{5})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{5})}$

Combine $4$ and $\frac{1}{5}$ .

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

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{4}))$

Multiply $\frac{5}{4}$ by $1$ .

$- \frac{5}{4}$

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.

$(5, \frac{15}{4})$

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

$x = 5$

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.

$y = \frac{25}{4}$

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

Direction: Opens Down

Vertex: $(5, 5)$

Focus: $(5, \frac{15}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{25}{4}$

Direction: Opens Down

Vertex: $(5, 5)$

Focus: $(5, \frac{15}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{25}{4}$

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 $4$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (4) = - \frac{16}{5} + 2 (4)$

Multiply $2$ by $4$ .

$f (4) = - \frac{16}{5} + 8$

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (4) = - \frac{16}{5} + 8 \cdot \frac{5}{5}$

Combine $8$ and $\frac{5}{5}$ .

$f (4) = - \frac{16}{5} + \frac{8 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (4) = \frac{- 16 + 8 \cdot 5}{5}$

Simplify the numerator.

Multiply $8$ by $5$ .

$f (4) = \frac{- 16 + 40}{5}$

Add $- 16$ and $40$ .

$f (4) = \frac{24}{5}$

The final answer is $\frac{24}{5}$ .

$\frac{24}{5}$

The $y$ value at $x = 4$ is $\frac{24}{5}$ .

$y = \frac{24}{5}$

Replace the variable $x$ with $3$ in the expression.

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

Simplify the result.

Simplify each term.

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

$f (3) = - \frac{9}{5} + 2 (3)$

Multiply $2$ by $3$ .

$f (3) = - \frac{9}{5} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (3) = - \frac{9}{5} + 6 \cdot \frac{5}{5}$

Combine $6$ and $\frac{5}{5}$ .

$f (3) = - \frac{9}{5} + \frac{6 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (3) = \frac{- 9 + 6 \cdot 5}{5}$

Simplify the numerator.

Multiply $6$ by $5$ .

$f (3) = \frac{- 9 + 30}{5}$

Add $- 9$ and $30$ .

$f (3) = \frac{21}{5}$

The final answer is $\frac{21}{5}$ .

$\frac{21}{5}$

The $y$ value at $x = 3$ is $\frac{21}{5}$ .

$y = \frac{21}{5}$

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

$f (6) = - \frac{{(6)}^{2}}{5} + 2 (6)$

Simplify the result.

Simplify each term.

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

$f (6) = - \frac{36}{5} + 2 (6)$

Multiply $2$ by $6$ .

$f (6) = - \frac{36}{5} + 12$

To write $12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (6) = - \frac{36}{5} + 12 \cdot \frac{5}{5}$

Combine $12$ and $\frac{5}{5}$ .

$f (6) = - \frac{36}{5} + \frac{12 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (6) = \frac{- 36 + 12 \cdot 5}{5}$

Simplify the numerator.

Multiply $12$ by $5$ .

$f (6) = \frac{- 36 + 60}{5}$

Add $- 36$ and $60$ .

$f (6) = \frac{24}{5}$

The final answer is $\frac{24}{5}$ .

$\frac{24}{5}$

The $y$ value at $x = 6$ is $\frac{24}{5}$ .

$y = \frac{24}{5}$

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

$f (7) = - \frac{{(7)}^{2}}{5} + 2 (7)$

Simplify the result.

Simplify each term.

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

$f (7) = - \frac{49}{5} + 2 (7)$

Multiply $2$ by $7$ .

$f (7) = - \frac{49}{5} + 14$

To write $14$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (7) = - \frac{49}{5} + 14 \cdot \frac{5}{5}$

Combine $14$ and $\frac{5}{5}$ .

$f (7) = - \frac{49}{5} + \frac{14 \cdot 5}{5}$

Combine the numerators over the common denominator.

$f (7) = \frac{- 49 + 14 \cdot 5}{5}$

Simplify the numerator.

Multiply $14$ by $5$ .

$f (7) = \frac{- 49 + 70}{5}$

Add $- 49$ and $70$ .

$f (7) = \frac{21}{5}$

The final answer is $\frac{21}{5}$ .

$\frac{21}{5}$

The $y$ value at $x = 7$ is $\frac{21}{5}$ .

$y = \frac{21}{5}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & \frac{21}{5} \\ 4 & \frac{24}{5} \\ 5 & 5 \\ 6 & \frac{24}{5} \\ 7 & \frac{21}{5} \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(5, 5)$

Focus: $(5, \frac{15}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{25}{4}$

$\begin{array}{c} x & y \\ 3 & \frac{21}{5} \\ 4 & \frac{24}{5} \\ 5 & 5 \\ 6 & \frac{24}{5} \\ 7 & \frac{21}{5} \end{array}$

Graph h(x)=-1/5x^2+2x

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