Graph f(x)=x-x^2

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

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

$a = - 1$

$b = 1$

$c = 0$

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

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

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

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

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

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

Simplify each term.

One to any power is one.

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

Multiply $4$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{1}{4}$ .

Multiply $- 1$ by $- 1$ .

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

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

$e = 0 + \frac{1}{4}$

Add $0$ and $\frac{1}{4}$ .

$e = \frac{1}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{1}{2})}^{2} + \frac{1}{4}$ .

$- {(x - \frac{1}{2})}^{2} + \frac{1}{4}$

Set $y$ equal to the new right side.

$y = - {(x - \frac{1}{2})}^{2} + \frac{1}{4}$

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

$a = - 1$

$h = \frac{1}{2}$

$k = \frac{1}{4}$

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

Opens Down

Find the vertex $(h, k)$ .

$(\frac{1}{2}, \frac{1}{4})$

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.

$(\frac{1}{2}, 0)$

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

$x = \frac{1}{2}$

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.

$y = \frac{1}{2}$

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

Direction: Opens Down

Vertex: $(\frac{1}{2}, \frac{1}{4})$

Focus: $(\frac{1}{2}, 0)$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = \frac{1}{2}$

Direction: Opens Down

Vertex: $(\frac{1}{2}, \frac{1}{4})$

Focus: $(\frac{1}{2}, 0)$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = \frac{1}{2}$

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 $- 1$ in the expression.

$f (- 1) = - {(- 1)}^{2} - 1$

Simplify the result.

Remove parentheses.

$f (- 1) = - {(- 1)}^{2} - 1$

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

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

$f (- 1) = (- 1) {(- 1)}^{2} - 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $2$ .

$f (- 1) = {(- 1)}^{3} - 1$

Raise $- 1$ to the power of $3$ .

$f (- 1) = - 1 - 1$

Subtract $1$ from $- 1$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$- 2$

The $y$ value at $x = - 1$ is $- 2$ .

$y = - 2$

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

$f (- 2) = - {(- 2)}^{2} - 2$

Simplify the result.

Remove parentheses.

$f (- 2) = - {(- 2)}^{2} - 2$

Simplify each term.

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

$f (- 2) = - 1 \cdot 4 - 2$

Multiply $- 1$ by $4$ .

$f (- 2) = - 4 - 2$

Subtract $2$ from $- 4$ .

$f (- 2) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = - 2$ is $- 6$ .

$y = - 6$

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

One to any power is one.

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

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 1$

Add $- 1$ and $1$ .

$f (1) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 1$ is $0$ .

$y = 0$

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

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

$f (2) = - 1 \cdot 4 + 2$

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 2$

Add $- 4$ and $2$ .

$f (2) = - 2$

The final answer is $- 2$ .

$- 2$

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

$y = - 2$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - 6 \\ - 1 & - 2 \\ \frac{1}{2} & \frac{1}{4} \\ 1 & 0 \\ 2 & - 2 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(\frac{1}{2}, \frac{1}{4})$

Focus: $(\frac{1}{2}, 0)$

Axis of Symmetry: $x = \frac{1}{2}$

Directrix: $y = \frac{1}{2}$

$\begin{array}{c} x & y \\ - 2 & - 6 \\ - 1 & - 2 \\ \frac{1}{2} & \frac{1}{4} \\ 1 & 0 \\ 2 & - 2 \end{array}$

Step 4

Graph f(x)=x-x^2

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