Graph y=1/2(x+2)^2-1

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

Find the properties of the given parabola.

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

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

$h = - 2$

$k = - 1$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 2, - 1)$

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.

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

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.

$y = - \frac{3}{2}$

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

Direction: Opens Up

Vertex: $(- 2, - 1)$

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

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{3}{2}$

Direction: Opens Up

Vertex: $(- 2, - 1)$

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

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{3}{2}$

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.

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

Simplify the result.

Simplify each term.

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

$f (- 4) = \frac{16}{2} + 2 (- 4) + 1$

Divide $16$ by $2$ .

$f (- 4) = 8 + 2 (- 4) + 1$

Multiply $2$ by $- 4$ .

$f (- 4) = 8 - 8 + 1$

Simplify by adding and subtracting.

Subtract $8$ from $8$ .

$f (- 4) = 0 + 1$

Add $0$ and $1$ .

$f (- 4) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = - 4$ is $1$ .

$y = 1$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $2$ by $- 3$ .

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

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 6$ by $2$ .

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

Subtract $12$ from $9$ .

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

Add $- 3$ and $2$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

The $y$ value at $x = - 3$ is $- \frac{1}{2}$ .

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

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

$f (0) = \frac{{(0)}^{2}}{2} + 2 (0) + 1$

Simplify the result.

Simplify each term.

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

$f (0) = \frac{0}{2} + 2 (0) + 1$

Divide $0$ by $2$ .

$f (0) = 0 + 2 (0) + 1$

Multiply $2$ by $0$ .

$f (0) = 0 + 0 + 1$

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 1$

Add $0$ and $1$ .

$f (0) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $2$ by $- 1$ .

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

Find the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $- 2$ by $2$ .

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

Subtract $4$ from $1$ .

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

Add $- 3$ and $2$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

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

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

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 2, - 1)$

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

Axis of Symmetry: $x = - 2$

Directrix: $y = - \frac{3}{2}$

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

Graph y=1/2(x+2)^2-1

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