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

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

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 $k$ .

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

$h = 4$

$k = - 2$

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

Opens Up

Find the vertex $(h, k)$ .

$(4, - 2)$

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.

$(4, - \frac{3}{2})$

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

$x = 4$

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: $(4, - 2)$

Focus: $(4, - \frac{3}{2})$

Axis of Symmetry: $x = 4$

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

Direction: Opens Up

Vertex: $(4, - 2)$

Focus: $(4, - \frac{3}{2})$

Axis of Symmetry: $x = 4$

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

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 $2$ in the expression.

$f (2) = \frac{{(2)}^{2}}{2} - 4 \cdot 2 + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of ${(2)}^{2}$ and $2$ .

Factor $2$ out of ${(2)}^{2}$ .

$f (2) = \frac{2 \cdot 2}{2} - 4 \cdot 2 + 6$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (2) = \frac{2 \cdot 2}{2 (1)} - 4 \cdot 2 + 6$

Cancel the common factor.

$f (2) = \frac{2 \cdot 2}{2 \cdot 1} - 4 \cdot 2 + 6$

Rewrite the expression.

$f (2) = \frac{2}{1} - 4 \cdot 2 + 6$

Divide $2$ by $1$ .

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

Multiply $- 4$ by $2$ .

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

Simplify by adding and subtracting.

Subtract $8$ from $2$ .

$f (2) = - 6 + 6$

Add $- 6$ and $6$ .

$f (2) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 2$ is $0$ .

$y = 0$

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

$f (3) = \frac{{(3)}^{2}}{2} - 4 \cdot 3 + 6$

Simplify the result.

Simplify each term.

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

$f (3) = \frac{9}{2} - 4 \cdot 3 + 6$

Multiply $- 4$ by $3$ .

$f (3) = \frac{9}{2} - 12 + 6$

Find the common denominator.

Write $- 12$ as a fraction with denominator $1$ .

$f (3) = \frac{9}{2} + \frac{- 12}{1} + 6$

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9}{2} + \frac{- 12}{1} \cdot \frac{2}{2} + 6$

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + 6$

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

$f (3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6}{1}$

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

$f (3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

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

$f (3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (3) = \frac{9 - 12 \cdot 2 + 6 \cdot 2}{2}$

Simplify each term.

Multiply $- 12$ by $2$ .

$f (3) = \frac{9 - 24 + 6 \cdot 2}{2}$

Multiply $6$ by $2$ .

$f (3) = \frac{9 - 24 + 12}{2}$

Simplify the expression.

Subtract $24$ from $9$ .

$f (3) = \frac{- 15 + 12}{2}$

Add $- 15$ and $12$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

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

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

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{2} - 4 \cdot 6 + 6$

Simplify the result.

Find the common denominator.

Write $- 4 \cdot 6$ as a fraction with denominator $1$ .

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6}{1} + 6$

Multiply $\frac{- 4 \cdot 6}{1}$ by $\frac{2}{2}$ .

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6}{1} \cdot \frac{2}{2} + 6$

Multiply $\frac{- 4 \cdot 6}{1}$ by $\frac{2}{2}$ .

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6 \cdot 2}{2} + 6$

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

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6 \cdot 2}{2} + \frac{6}{1}$

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

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

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

$f (6) = \frac{{(6)}^{2}}{2} + \frac{- 4 \cdot 6 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (6) = \frac{{(6)}^{2} - 4 \cdot 6 \cdot 2 + 6 \cdot 2}{2}$

Simplify each term.

Raise $6$ to the power of $2$ .

$f (6) = \frac{36 - 4 \cdot 6 \cdot 2 + 6 \cdot 2}{2}$

Multiply $- 4$ by $6$ .

$f (6) = \frac{36 - 24 \cdot 2 + 6 \cdot 2}{2}$

Multiply $- 24$ by $2$ .

$f (6) = \frac{36 - 48 + 6 \cdot 2}{2}$

Multiply $6$ by $2$ .

$f (6) = \frac{36 - 48 + 12}{2}$

Simplify the expression.

Subtract $48$ from $36$ .

$f (6) = \frac{- 12 + 12}{2}$

Add $- 12$ and $12$ .

$f (6) = \frac{0}{2}$

Divide $0$ by $2$ .

$f (6) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 6$ is $0$ .

$y = 0$

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{{(5)}^{2}}{2} - 4 \cdot 5 + 6$

Simplify the result.

Simplify each term.

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

$f (5) = \frac{25}{2} - 4 \cdot 5 + 6$

Multiply $- 4$ by $5$ .

$f (5) = \frac{25}{2} - 20 + 6$

Find the common denominator.

Write $- 20$ as a fraction with denominator $1$ .

$f (5) = \frac{25}{2} + \frac{- 20}{1} + 6$

Multiply $\frac{- 20}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{25}{2} + \frac{- 20}{1} \cdot \frac{2}{2} + 6$

Multiply $\frac{- 20}{1}$ by $\frac{2}{2}$ .

$f (5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + 6$

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

$f (5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6}{1}$

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

$f (5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

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

$f (5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (5) = \frac{25 - 20 \cdot 2 + 6 \cdot 2}{2}$

Simplify each term.

Multiply $- 20$ by $2$ .

$f (5) = \frac{25 - 40 + 6 \cdot 2}{2}$

Multiply $6$ by $2$ .

$f (5) = \frac{25 - 40 + 12}{2}$

Simplify the expression.

Subtract $40$ from $25$ .

$f (5) = \frac{- 15 + 12}{2}$

Add $- 15$ and $12$ .

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

Move the negative in front of the fraction.

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

The final answer is $- \frac{3}{2}$ .

$- \frac{3}{2}$

The $y$ value at $x = 5$ is $- \frac{3}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 0 \\ 3 & - \frac{3}{2} \\ 4 & - 2 \\ 5 & - \frac{3}{2} \\ 6 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 2)$

Focus: $(4, - \frac{3}{2})$

Axis of Symmetry: $x = 4$

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

$\begin{array}{c} x & y \\ 2 & 0 \\ 3 & - \frac{3}{2} \\ 4 & - 2 \\ 5 & - \frac{3}{2} \\ 6 & 0 \end{array}$

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

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