Graph y=1/2(x+1)^2+3

$y = \frac{1}{2} {(x + 1)}^{2} + 3$

Step 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 $k$ .

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

$h = - 1$

$k = 3$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 1, 3)$

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.

$(- 1, \frac{7}{2})$

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.

$y = \frac{5}{2}$

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

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{5}{2}$

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{5}{2}$

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 $- 3$ in the expression.

$f (- 3) = \frac{{(- 3)}^{2}}{2} - 3 + \frac{7}{2}$

Simplify the result.

Remove parentheses.

$f (- 3) = \frac{{(- 3)}^{2}}{2} - 3 + \frac{7}{2}$

Combine the numerators over the common denominator.

$f (- 3) = - 3 + \frac{{(- 3)}^{2} + 7}{2}$

Simplify the expression.

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

$f (- 3) = - 3 + \frac{9 + 7}{2}$

Add $9$ and $7$ .

$f (- 3) = - 3 + \frac{16}{2}$

Divide $16$ by $2$ .

$f (- 3) = - 3 + 8$

Add $- 3$ and $8$ .

$f (- 3) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = - 3$ is $5$ .

$y = 5$

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

$f (- 2) = \frac{{(- 2)}^{2}}{2} - 2 + \frac{7}{2}$

Simplify the result.

Remove parentheses.

$f (- 2) = \frac{{(- 2)}^{2}}{2} - 2 + \frac{7}{2}$

Combine the numerators over the common denominator.

$f (- 2) = - 2 + \frac{{(- 2)}^{2} + 7}{2}$

Simplify the expression.

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

$f (- 2) = - 2 + \frac{4 + 7}{2}$

Add $4$ and $7$ .

$f (- 2) = - 2 + \frac{11}{2}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - 2 \cdot \frac{2}{2} + \frac{11}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$f (- 2) = \frac{- 2 \cdot 2}{2} + \frac{11}{2}$

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 2 \cdot 2 + 11}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$f (- 2) = \frac{- 4 + 11}{2}$

Add $- 4$ and $11$ .

$f (- 2) = \frac{7}{2}$

The final answer is $\frac{7}{2}$ .

$\frac{7}{2}$

The $y$ value at $x = - 2$ is $\frac{7}{2}$ .

$y = \frac{7}{2}$

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

$f (1) = \frac{{(1)}^{2}}{2} + 1 + \frac{7}{2}$

Simplify the result.

Remove parentheses.

$f (1) = \frac{{(1)}^{2}}{2} + 1 + \frac{7}{2}$

Combine the numerators over the common denominator.

$f (1) = 1 + \frac{{(1)}^{2} + 7}{2}$

Simplify the expression.

One to any power is one.

$f (1) = 1 + \frac{1 + 7}{2}$

Add $1$ and $7$ .

$f (1) = 1 + \frac{8}{2}$

Divide $8$ by $2$ .

$f (1) = 1 + 4$

Add $1$ and $4$ .

$f (1) = 5$

The final answer is $5$ .

$5$

The $y$ value at $x = 1$ is $5$ .

$y = 5$

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

$f (0) = \frac{{(0)}^{2}}{2} + 0 + \frac{7}{2}$

Simplify the result.

Remove parentheses.

$f (0) = \frac{{(0)}^{2}}{2} + 0 + \frac{7}{2}$

Combine the numerators over the common denominator.

$f (0) = \frac{{(0)}^{2} + 7}{2}$

Simplify the expression.

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

$f (0) = \frac{0 + 7}{2}$

Add $0$ and $7$ .

$f (0) = \frac{7}{2}$

The final answer is $\frac{7}{2}$ .

$\frac{7}{2}$

The $y$ value at $x = 0$ is $\frac{7}{2}$ .

$y = \frac{7}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 5 \\ - 2 & \frac{7}{2} \\ - 1 & 3 \\ 0 & \frac{7}{2} \\ 1 & 5 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{7}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{5}{2}$

$\begin{array}{c} x & y \\ - 3 & 5 \\ - 2 & \frac{7}{2} \\ - 1 & 3 \\ 0 & \frac{7}{2} \\ 1 & 5 \end{array}$

Step 4

Graph y=1/2(x+1)^2+3

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