Graph y=-1/2(x+3)^2+4

$y = - \frac{1}{2} {(x + 3)}^{2} + 4$

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 $k$ .

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

$h = - 3$

$k = 4$

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Find the vertex $(h, k)$ .

$(- 3, 4)$

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.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

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.

$(- 3, \frac{7}{2})$

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 Down

Vertex: $(- 3, 4)$

Focus: $(- 3, \frac{7}{2})$

Axis of Symmetry: $x = - 3$

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

Direction: Opens Down

Vertex: $(- 3, 4)$

Focus: $(- 3, \frac{7}{2})$

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.

$f (- 5) = - \frac{{(- 5)}^{2}}{2} - 3 \cdot - 5 - \frac{1}{2}$

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (- 5) = - 3 \cdot - 5 + \frac{- 1 \cdot 25 - 1}{2}$

Multiply $- 1$ by $25$ .

$f (- 5) = - 3 \cdot - 5 + \frac{- 25 - 1}{2}$

Subtract $1$ from $- 25$ .

$f (- 5) = - 3 \cdot - 5 + \frac{- 26}{2}$

Simplify each term.

Multiply $- 3$ by $- 5$ .

$f (- 5) = 15 + \frac{- 26}{2}$

Divide $- 26$ by $2$ .

$f (- 5) = 15 - 13$

Subtract $13$ from $15$ .

$f (- 5) = 2$

The final answer is $2$ .

$2$

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

$y = 2$

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

$f (- 4) = - \frac{{(- 4)}^{2}}{2} - 3 \cdot - 4 - \frac{1}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (- 4) = - 3 \cdot - 4 + \frac{- {(- 4)}^{2} - 1}{2}$

Simplify each term.

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

$f (- 4) = - 3 \cdot - 4 + \frac{- 1 \cdot 16 - 1}{2}$

Multiply $- 1$ by $16$ .

$f (- 4) = - 3 \cdot - 4 + \frac{- 16 - 1}{2}$

Subtract $1$ from $- 16$ .

$f (- 4) = - 3 \cdot - 4 + \frac{- 17}{2}$

Simplify each term.

Multiply $- 3$ by $- 4$ .

$f (- 4) = 12 + \frac{- 17}{2}$

Move the negative in front of the fraction.

$f (- 4) = 12 - \frac{17}{2}$

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

$f (- 4) = 12 \cdot \frac{2}{2} - \frac{17}{2}$

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

$f (- 4) = \frac{12 \cdot 2}{2} - \frac{17}{2}$

Combine the numerators over the common denominator.

$f (- 4) = \frac{12 \cdot 2 - 17}{2}$

Simplify the numerator.

Multiply $12$ by $2$ .

$f (- 4) = \frac{24 - 17}{2}$

Subtract $17$ from $24$ .

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

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

$\frac{7}{2}$

The $y$ value at $x = - 4$ is $\frac{7}{2}$ .

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

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

Raise $- 1$ to the power of $3$ .

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

Subtract $1$ from $- 1$ .

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

Simplify each term.

Multiply $- 3$ by $- 1$ .

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

Divide $- 2$ by $2$ .

$f (- 1) = 3 - 1$

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.

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (- 2) = - 3 \cdot - 2 + \frac{- 1 \cdot 4 - 1}{2}$

Multiply $- 1$ by $4$ .

$f (- 2) = - 3 \cdot - 2 + \frac{- 4 - 1}{2}$

Subtract $1$ from $- 4$ .

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

Simplify each term.

Multiply $- 3$ by $- 2$ .

$f (- 2) = 6 + \frac{- 5}{2}$

Move the negative in front of the fraction.

$f (- 2) = 6 - \frac{5}{2}$

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

$f (- 2) = 6 \cdot \frac{2}{2} - \frac{5}{2}$

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

$f (- 2) = \frac{6 \cdot 2}{2} - \frac{5}{2}$

Combine the numerators over the common denominator.

$f (- 2) = \frac{6 \cdot 2 - 5}{2}$

Simplify the numerator.

Multiply $6$ by $2$ .

$f (- 2) = \frac{12 - 5}{2}$

Subtract $5$ from $12$ .

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

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

$\frac{7}{2}$

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

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

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 3, 4)$

Focus: $(- 3, \frac{7}{2})$

Axis of Symmetry: $x = - 3$

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

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

Graph y=-1/2(x+3)^2+4

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