Graph f(x)=-1/2(x-2)^2-4

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

Step 1

Find the properties of the given parabola.

Isolate $y$ to the left side of the equation.

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

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

Reorder terms.

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

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

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

$h = 2$

$k = - 4$

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

Opens Down

Find the vertex $(h, k)$ .

$(2, - 4)$

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.

$(2, - \frac{9}{2})$

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.

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

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

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{9}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{7}{2}$

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{9}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{7}{2}$

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 $0$ 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 $- 1$ by $0$ .

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

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.

One to any power is one.

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

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.

Add $- 1$ and $4$ .

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

Subtract $12$ from $3$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

The $y$ value at $x = 1$ is $- \frac{9}{2}$ .

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

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 $2$ .

$f (4) = - 1 \cdot 8 + 2 (4) - 6$

Multiply $- 1$ by $8$ .

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

Multiply $2$ by $4$ .

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

Simplify by adding and subtracting.

Add $- 8$ and $8$ .

$f (4) = 0 - 6$

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.

Add $- 9$ and $12$ .

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

Subtract $12$ from $3$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

The $y$ value at $x = 3$ is $- \frac{9}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 6 \\ 1 & - \frac{9}{2} \\ 2 & - 4 \\ 3 & - \frac{9}{2} \\ 4 & - 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(2, - 4)$

Focus: $(2, - \frac{9}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{7}{2}$

$\begin{array}{c} x & y \\ 0 & - 6 \\ 1 & - \frac{9}{2} \\ 2 & - 4 \\ 3 & - \frac{9}{2} \\ 4 & - 6 \end{array}$

Step 4

Graph f(x)=-1/2(x-2)^2-4

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