Graph f(x)=2x^2-x-1

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} - x - 1$ .

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

$a = 2$

$b = - 1$

$c = - 1$

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

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

Simplify the right side.

Multiply $2$ by $2$ .

$d = \frac{- 1}{4}$

Move the negative in front of the fraction.

$d = - \frac{1}{4}$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $2$ .

$e = - 1 - \frac{1}{8}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 1 \cdot \frac{8}{8} - \frac{1}{8}$

Combine $- 1$ and $\frac{8}{8}$ .

$e = \frac{- 1 \cdot 8}{8} - \frac{1}{8}$

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 8 - 1}{8}$

Simplify the numerator.

Multiply $- 1$ by $8$ .

$e = \frac{- 8 - 1}{8}$

Subtract $1$ from $- 8$ .

$e = \frac{- 9}{8}$

Move the negative in front of the fraction.

$e = - \frac{9}{8}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - \frac{1}{4})}^{2} - \frac{9}{8}$ .

$2 {(x - \frac{1}{4})}^{2} - \frac{9}{8}$

Set $y$ equal to the new right side.

$y = 2 {(x - \frac{1}{4})}^{2} - \frac{9}{8}$

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

$a = 2$

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

$k = - \frac{9}{8}$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(\frac{1}{4}, - \frac{9}{8})$

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.

$\frac{1}{4 \cdot 2}$

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

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

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

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

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.

$y = - \frac{5}{4}$

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

Direction: Opens Up

Vertex: $(\frac{1}{4}, - \frac{9}{8})$

Focus: $(\frac{1}{4}, - 1)$

Axis of Symmetry: $x = \frac{1}{4}$

Directrix: $y = - \frac{5}{4}$

Direction: Opens Up

Vertex: $(\frac{1}{4}, - \frac{9}{8})$

Focus: $(\frac{1}{4}, - 1)$

Axis of Symmetry: $x = \frac{1}{4}$

Directrix: $y = - \frac{5}{4}$

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 $- 1$ in the expression.

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

Simplify the result.

Simplify each term.

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

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

Multiply $2$ by $1$ .

$f (- 1) = 2 - (- 1) - 1$

Multiply $- 1$ by $- 1$ .

$f (- 1) = 2 + 1 - 1$

Simplify by adding and subtracting.

Add $2$ and $1$ .

$f (- 1) = 3 - 1$

Subtract $1$ from $3$ .

$f (- 1) = 2$

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.

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

Simplify the result.

Simplify each term.

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

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

Multiply $2$ by $4$ .

$f (- 2) = 8 - (- 2) - 1$

Multiply $- 1$ by $- 2$ .

$f (- 2) = 8 + 2 - 1$

Simplify by adding and subtracting.

Add $8$ and $2$ .

$f (- 2) = 10 - 1$

Subtract $1$ from $10$ .

$f (- 2) = 9$

The final answer is $9$ .

$9$

The $y$ value at $x = - 2$ is $9$ .

$y = 9$

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

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

Simplify the result.

Simplify each term.

One to any power is one.

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

Multiply $2$ by $1$ .

$f (1) = 2 - (1) - 1$

Multiply $- 1$ by $1$ .

$f (1) = 2 - 1 - 1$

Simplify by subtracting numbers.

Subtract $1$ from $2$ .

$f (1) = 1 - 1$

Subtract $1$ from $1$ .

$f (1) = 0$

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.

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

Simplify the result.

Simplify each term.

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

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

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

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

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

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

Add $1$ and $2$ .

$f (2) = 2^{3} - (2) - 1$

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

$f (2) = 8 - (2) - 1$

Multiply $- 1$ by $2$ .

$f (2) = 8 - 2 - 1$

Simplify by subtracting numbers.

Subtract $2$ from $8$ .

$f (2) = 6 - 1$

Subtract $1$ from $6$ .

$f (2) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = 2$ is $5$ .

$y = 5$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 9 \\ - 1 & 2 \\ \frac{1}{4} & - \frac{9}{8} \\ 1 & 0 \\ 2 & 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{1}{4}, - \frac{9}{8})$

Focus: $(\frac{1}{4}, - 1)$

Axis of Symmetry: $x = \frac{1}{4}$

Directrix: $y = - \frac{5}{4}$

$\begin{array}{c} x & y \\ - 2 & 9 \\ - 1 & 2 \\ \frac{1}{4} & - \frac{9}{8} \\ 1 & 0 \\ 2 & 5 \end{array}$

Step 4

Graph f(x)=2x^2-x-1

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