Graph y-5=1/2(x+3)^2

$y - 5 = \frac{1}{2} {(x + 3)}^{2}$

Solve for $y$ .

Simplify $\frac{1}{2} \cdot {(x + 3)}^{2}$ .

Rewrite.

$y - 5 = 0 + 0 + \frac{1}{2} \cdot {(x + 3)}^{2}$

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y - 5 = \frac{1}{2} \cdot ((x + 3) (x + 3))$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$y - 5 = \frac{1}{2} \cdot (x (x + 3) + 3 (x + 3))$

Apply the distributive property.

$y - 5 = \frac{1}{2} \cdot (x \cdot x + x \cdot 3 + 3 (x + 3))$

Apply the distributive property.

$y - 5 = \frac{1}{2} \cdot (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y - 5 = \frac{1}{2} \cdot (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Move $3$ to the left of $x$ .

$y - 5 = \frac{1}{2} \cdot (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Multiply $3$ by $3$ .

$y - 5 = \frac{1}{2} \cdot (x^{2} + 3 x + 3 x + 9)$

Add $3 x$ and $3 x$ .

$y - 5 = \frac{1}{2} \cdot (x^{2} + 6 x + 9)$

Apply the distributive property.

$y - 5 = \frac{1}{2} x^{2} + \frac{1}{2} (6 x) + \frac{1}{2} \cdot 9$

Simplify.

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

$y - 5 = \frac{x^{2}}{2} + \frac{1}{2} (6 x) + \frac{1}{2} \cdot 9$

Cancel the common factor of $2$ .

Factor $2$ out of $6 x$ .

$y - 5 = \frac{x^{2}}{2} + \frac{1}{2} (2 (3 x)) + \frac{1}{2} \cdot 9$

Cancel the common factor.

$y - 5 = \frac{x^{2}}{2} + \frac{1}{2} (2 (3 x)) + \frac{1}{2} \cdot 9$

Rewrite the expression.

$y - 5 = \frac{x^{2}}{2} + 3 x + \frac{1}{2} \cdot 9$

Combine $\frac{1}{2}$ and $9$ .

$y - 5 = \frac{x^{2}}{2} + 3 x + \frac{9}{2}$

Move all terms not containing $y$ to the right side of the equation.

Add $5$ to both sides of the equation.

$y = \frac{x^{2}}{2} + 3 x + \frac{9}{2} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x^{2}}{2} + 3 x + \frac{9}{2} + 5 \cdot \frac{2}{2}$

Combine $5$ and $\frac{2}{2}$ .

$y = \frac{x^{2}}{2} + 3 x + \frac{9}{2} + \frac{5 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{2} + 3 x + \frac{9 + 5 \cdot 2}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$y = \frac{x^{2}}{2} + 3 x + \frac{9 + 10}{2}$

Add $9$ and $10$ .

$y = \frac{x^{2}}{2} + 3 x + \frac{19}{2}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{x^{2}}{2} + 3 x + \frac{19}{2}$ .

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

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

$b = 3$

$c = \frac{19}{2}$

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

$d = \frac{3}{2 (\frac{1}{2})}$

Simplify the right side.

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

$d = \frac{3}{\frac{2}{2}}$

Divide $2$ by $2$ .

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = \frac{19}{2} - \frac{3^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = \frac{19}{2} - \frac{9}{4 (\frac{1}{2})}$

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

$e = \frac{19}{2} - \frac{9}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = \frac{19}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$e = \frac{19 - 9}{2}$

Subtract $9$ from $19$ .

$e = \frac{10}{2}$

Divide $10$ by $2$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x + 3)}^{2} + 5$ .

$\frac{1}{2} {(x + 3)}^{2} + 5$

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 3)}^{2} + 5$

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

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

$h = - 3$

$k = 5$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 3, 5)$

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.

$(- 3, \frac{11}{2})$

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

$x = - 3$

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.

$y = \frac{9}{2}$

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

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, \frac{11}{2})$

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{9}{2}$

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, \frac{11}{2})$

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{9}{2}$

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 $- 5$ in the expression.

$f (- 5) = \frac{{(- 5)}^{2}}{2} + 3 (- 5) + \frac{19}{2}$

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

$f (- 5) = 3 (- 5) + \frac{{(- 5)}^{2} + 19}{2}$

Simplify the expression.

Raise $- 5$ to the power of $2$ .

$f (- 5) = 3 (- 5) + \frac{25 + 19}{2}$

Add $25$ and $19$ .

$f (- 5) = 3 (- 5) + \frac{44}{2}$

Simplify each term.

Multiply $3$ by $- 5$ .

$f (- 5) = - 15 + \frac{44}{2}$

Divide $44$ by $2$ .

$f (- 5) = - 15 + 22$

Add $- 15$ and $22$ .

$f (- 5) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = - 5$ is $7$ .

$y = 7$

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{{(- 4)}^{2}}{2} + 3 (- 4) + \frac{19}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 4) = 3 (- 4) + \frac{{(- 4)}^{2} + 19}{2}$

Simplify the expression.

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

$f (- 4) = 3 (- 4) + \frac{16 + 19}{2}$

Add $16$ and $19$ .

$f (- 4) = 3 (- 4) + \frac{35}{2}$

Multiply $3$ by $- 4$ .

$f (- 4) = - 12 + \frac{35}{2}$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 4) = - 12 \cdot \frac{2}{2} + \frac{35}{2}$

Combine $- 12$ and $\frac{2}{2}$ .

$f (- 4) = \frac{- 12 \cdot 2}{2} + \frac{35}{2}$

Combine the numerators over the common denominator.

$f (- 4) = \frac{- 12 \cdot 2 + 35}{2}$

Simplify the numerator.

Multiply $- 12$ by $2$ .

$f (- 4) = \frac{- 24 + 35}{2}$

Add $- 24$ and $35$ .

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

The final answer is $\frac{11}{2}$ .

$\frac{11}{2}$

The $y$ value at $x = - 4$ is $\frac{11}{2}$ .

$y = \frac{11}{2}$

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

$f (- 1) = \frac{{(- 1)}^{2}}{2} + 3 (- 1) + \frac{19}{2}$

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

$f (- 1) = 3 (- 1) + \frac{{(- 1)}^{2} + 19}{2}$

Simplify the expression.

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

$f (- 1) = 3 (- 1) + \frac{1 + 19}{2}$

Add $1$ and $19$ .

$f (- 1) = 3 (- 1) + \frac{20}{2}$

Simplify each term.

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 + \frac{20}{2}$

Divide $20$ by $2$ .

$f (- 1) = - 3 + 10$

Add $- 3$ and $10$ .

$f (- 1) = 7$

The final answer is $7$ .

$7$

The $y$ value at $x = - 1$ is $7$ .

$y = 7$

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

$f (- 2) = \frac{{(- 2)}^{2}}{2} + 3 (- 2) + \frac{19}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 2) = 3 (- 2) + \frac{{(- 2)}^{2} + 19}{2}$

Simplify the expression.

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

$f (- 2) = 3 (- 2) + \frac{4 + 19}{2}$

Add $4$ and $19$ .

$f (- 2) = 3 (- 2) + \frac{23}{2}$

Multiply $3$ by $- 2$ .

$f (- 2) = - 6 + \frac{23}{2}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - 6 \cdot \frac{2}{2} + \frac{23}{2}$

Combine $- 6$ and $\frac{2}{2}$ .

$f (- 2) = \frac{- 6 \cdot 2}{2} + \frac{23}{2}$

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 6 \cdot 2 + 23}{2}$

Simplify the numerator.

Multiply $- 6$ by $2$ .

$f (- 2) = \frac{- 12 + 23}{2}$

Add $- 12$ and $23$ .

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

The final answer is $\frac{11}{2}$ .

$\frac{11}{2}$

The $y$ value at $x = - 2$ is $\frac{11}{2}$ .

$y = \frac{11}{2}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & 7 \\ - 4 & \frac{11}{2} \\ - 3 & 5 \\ - 2 & \frac{11}{2} \\ - 1 & 7 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, \frac{11}{2})$

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{9}{2}$

$\begin{array}{c} x & y \\ - 5 & 7 \\ - 4 & \frac{11}{2} \\ - 3 & 5 \\ - 2 & \frac{11}{2} \\ - 1 & 7 \end{array}$

Graph y-5=1/2(x+3)^2

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