Graph f(x)=1/2x^2+2x-6

$f (x) = \frac{1}{2} x^{2} + 2 x - 6$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

$y = \frac{x^{2}}{2} + 2 x - 6$

Complete the square for $\frac{x^{2}}{2} + 2 x - 6$ .

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

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

$b = 2$

$c = - 6$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Multiply $2$ by $1$ .

$d = 2$

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

$e = - 6 - \frac{2^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = - 6 - \frac{4}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$e = - 6 - \frac{4}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = - 6 - \frac{4}{2}$

Divide $4$ by $2$ .

$e = - 6 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = - 6 - 2$

Subtract $2$ from $- 6$ .

$e = - 8$

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

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

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 2)}^{2} - 8$

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

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

$h = - 2$

$k = - 8$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 2, - 8)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{2}$ .

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

Divide $4$ by $2$ .

$\frac{1}{2}$

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.

$(- 2, - \frac{15}{2})$

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

$x = - 2$

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.

$y = - \frac{17}{2}$

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

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{15}{2})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{17}{2}$

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{15}{2})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{17}{2}$

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 $- 4$ in the expression.

$f (- 4) = \frac{{(- 4)}^{2}}{2} + 2 (- 4) - 6$

Simplify the result.

Simplify each term.

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

$f (- 4) = \frac{16}{2} + 2 (- 4) - 6$

Divide $16$ by $2$ .

$f (- 4) = 8 + 2 (- 4) - 6$

Multiply $2$ by $- 4$ .

$f (- 4) = 8 - 8 - 6$

Simplify by subtracting numbers.

Subtract $8$ from $8$ .

$f (- 4) = 0 - 6$

Subtract $6$ from $0$ .

$f (- 4) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = - 4$ is $- 6$ .

$y = - 6$

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

$f (- 3) = \frac{{(- 3)}^{2}}{2} + 2 (- 3) - 6$

Simplify the result.

Simplify each term.

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

$f (- 3) = \frac{9}{2} + 2 (- 3) - 6$

Multiply $2$ by $- 3$ .

$f (- 3) = \frac{9}{2} - 6 - 6$

Find the common denominator.

Write $- 6$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{2} + \frac{- 6}{1} - 6$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 6}{1} \cdot \frac{2}{2} - 6$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 6 \cdot 2}{2} - 6$

Write $- 6$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{2} + \frac{- 6 \cdot 2}{2} + \frac{- 6}{1}$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 6 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 6 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 - 6 \cdot 2 - 6 \cdot 2}{2}$

Simplify each term.

Multiply $- 6$ by $2$ .

$f (- 3) = \frac{9 - 12 - 6 \cdot 2}{2}$

Multiply $- 6$ by $2$ .

$f (- 3) = \frac{9 - 12 - 12}{2}$

Simplify the expression.

Subtract $12$ from $9$ .

$f (- 3) = \frac{- 3 - 12}{2}$

Subtract $12$ from $- 3$ .

$f (- 3) = \frac{- 15}{2}$

Move the negative in front of the fraction.

$f (- 3) = - \frac{15}{2}$

The final answer is $- \frac{15}{2}$ .

$- \frac{15}{2}$

The $y$ value at $x = - 3$ is $- \frac{15}{2}$ .

$y = - \frac{15}{2}$

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{2}}{2} + 2 (0) - 6$

Simplify the result.

Simplify each term.

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

$f (0) = \frac{0}{2} + 2 (0) - 6$

Divide $0$ by $2$ .

$f (0) = 0 + 2 (0) - 6$

Multiply $2$ by $0$ .

$f (0) = 0 + 0 - 6$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 6$

Subtract $6$ from $0$ .

$f (0) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = 0$ is $- 6$ .

$y = - 6$

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

$f (- 1) = \frac{{(- 1)}^{2}}{2} + 2 (- 1) - 6$

Simplify the result.

Simplify each term.

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

$f (- 1) = \frac{1}{2} + 2 (- 1) - 6$

Multiply $2$ by $- 1$ .

$f (- 1) = \frac{1}{2} - 2 - 6$

Find the common denominator.

Write $- 2$ as a fraction with denominator $1$ .

$f (- 1) = \frac{1}{2} + \frac{- 2}{1} - 6$

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{1}{2} + \frac{- 2}{1} \cdot \frac{2}{2} - 6$

Multiply $\frac{- 2}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{1}{2} + \frac{- 2 \cdot 2}{2} - 6$

Write $- 6$ as a fraction with denominator $1$ .

$f (- 1) = \frac{1}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6}{1}$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{1}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6}{1} \cdot \frac{2}{2}$

Multiply $\frac{- 6}{1}$ by $\frac{2}{2}$ .

$f (- 1) = \frac{1}{2} + \frac{- 2 \cdot 2}{2} + \frac{- 6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{1 - 2 \cdot 2 - 6 \cdot 2}{2}$

Simplify each term.

Multiply $- 2$ by $2$ .

$f (- 1) = \frac{1 - 4 - 6 \cdot 2}{2}$

Multiply $- 6$ by $2$ .

$f (- 1) = \frac{1 - 4 - 12}{2}$

Simplify the expression.

Subtract $4$ from $1$ .

$f (- 1) = \frac{- 3 - 12}{2}$

Subtract $12$ from $- 3$ .

$f (- 1) = \frac{- 15}{2}$

Move the negative in front of the fraction.

$f (- 1) = - \frac{15}{2}$

The final answer is $- \frac{15}{2}$ .

$- \frac{15}{2}$

The $y$ value at $x = - 1$ is $- \frac{15}{2}$ .

$y = - \frac{15}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & - 6 \\ - 3 & - \frac{15}{2} \\ - 2 & - 8 \\ - 1 & - \frac{15}{2} \\ 0 & - 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 2, - 8)$

Focus: $(- 2, - \frac{15}{2})$

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{17}{2}$

$\begin{array}{c} x & y \\ - 4 & - 6 \\ - 3 & - \frac{15}{2} \\ - 2 & - 8 \\ - 1 & - \frac{15}{2} \\ 0 & - 6 \end{array}$

Step 4

Graph f(x)=1/2x^2+2x-6

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