Graph f(x)=10x-5x^2

$f (x) = 10 x - 5 x^{2}$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Reorder $10 x$ and $- 5 x^{2}$ .

$y = - 5 x^{2} + 10 x$

Complete the square for $- 5 x^{2} + 10 x$ .

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

$a = - 5$

$b = 10$

$c = 0$

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

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

Simplify the right side.

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5$ and $- 5$ .

Factor $5$ out of $5$ .

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

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

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{10^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = 0 - \frac{100}{- 20}$

Divide $100$ by $- 20$ .

$e = 0 - - 5$

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

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

Set $y$ equal to the new right side.

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

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

$a = - 5$

$h = 1$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(1, 5)$

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.

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

Simplify.

Multiply $4$ by $- 5$ .

$\frac{1}{- 20}$

Move the negative in front of the fraction.

$- \frac{1}{20}$

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.

$(1, \frac{99}{20})$

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

$x = 1$

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.

$y = \frac{101}{20}$

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

Direction: Opens Down

Vertex: $(1, 5)$

Focus: $(1, \frac{99}{20})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{101}{20}$

Direction: Opens Down

Vertex: $(1, 5)$

Focus: $(1, \frac{99}{20})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{101}{20}$

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 $0$ in the expression.

$f (0) = - 5 {(0)}^{2} + 10 (0)$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = - 5 \cdot 0 + 10 (0)$

Multiply $- 5$ by $0$ .

$f (0) = 0 + 10 (0)$

Multiply $10$ by $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 0$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - 5 {(- 1)}^{2} + 10 (- 1)$

Simplify the result.

Simplify each term.

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

$f (- 1) = - 5 \cdot 1 + 10 (- 1)$

Multiply $- 5$ by $1$ .

$f (- 1) = - 5 + 10 (- 1)$

Multiply $10$ by $- 1$ .

$f (- 1) = - 5 - 10$

Subtract $10$ from $- 5$ .

$f (- 1) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = - 1$ is $- 15$ .

$y = - 15$

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

$f (2) = - 5 {(2)}^{2} + 10 (2)$

Simplify the result.

Simplify each term.

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

$f (2) = - 5 \cdot 4 + 10 (2)$

Multiply $- 5$ by $4$ .

$f (2) = - 20 + 10 (2)$

Multiply $10$ by $2$ .

$f (2) = - 20 + 20$

Add $- 20$ and $20$ .

$f (2) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 2$ is $0$ .

$y = 0$

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

$f (3) = - 5 {(3)}^{2} + 10 (3)$

Simplify the result.

Simplify each term.

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

$f (3) = - 5 \cdot 9 + 10 (3)$

Multiply $- 5$ by $9$ .

$f (3) = - 45 + 10 (3)$

Multiply $10$ by $3$ .

$f (3) = - 45 + 30$

Add $- 45$ and $30$ .

$f (3) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = 3$ is $- 15$ .

$y = - 15$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 15 \\ 0 & 0 \\ 1 & 5 \\ 2 & 0 \\ 3 & - 15 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, 5)$

Focus: $(1, \frac{99}{20})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{101}{20}$

$\begin{array}{c} x & y \\ - 1 & - 15 \\ 0 & 0 \\ 1 & 5 \\ 2 & 0 \\ 3 & - 15 \end{array}$

Step 4

Graph f(x)=10x-5x^2

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