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

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

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 $k$ .

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

$h = 3$

$k = - 2$

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

Opens Up

Find the vertex $(h, k)$ .

$(3, - 2)$

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.

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

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.

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

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

Direction: Opens Up

Vertex: $(3, - 2)$

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

Axis of Symmetry: $x = 3$

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

Direction: Opens Up

Vertex: $(3, - 2)$

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

Axis of Symmetry: $x = 3$

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 $1$ in the expression.

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

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

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

Simplify the expression.

One to any power is one.

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

Add $1$ and $5$ .

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

Simplify each term.

Multiply $- 3$ by $1$ .

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

Divide $6$ by $2$ .

$f (1) = - 3 + 3$

Add $- 3$ and $3$ .

$f (1) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 1$ is $0$ .

$y = 0$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify the expression.

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

$f (2) = - 3 \cdot 2 + \frac{4 + 5}{2}$

Add $4$ and $5$ .

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

Multiply $- 3$ by $2$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 6$ by $2$ .

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

Add $- 12$ and $9$ .

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

Move the negative in front of the fraction.

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

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

$- \frac{3}{2}$

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

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

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

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

Simplify the result.

Combine fractions.

Combine the numerators over the common denominator.

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

Simplify the expression.

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

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

Add $25$ and $5$ .

$f (5) = - 3 \cdot 5 + \frac{30}{2}$

Simplify each term.

Multiply $- 3$ by $5$ .

$f (5) = - 15 + \frac{30}{2}$

Divide $30$ by $2$ .

$f (5) = - 15 + 15$

Add $- 15$ and $15$ .

$f (5) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 5$ is $0$ .

$y = 0$

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

$f (4) = \frac{{(4)}^{2}}{2} - 3 \cdot 4 + \frac{5}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (4) = - 3 \cdot 4 + \frac{{(4)}^{2} + 5}{2}$

Simplify the expression.

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

$f (4) = - 3 \cdot 4 + \frac{16 + 5}{2}$

Add $16$ and $5$ .

$f (4) = - 3 \cdot 4 + \frac{21}{2}$

Multiply $- 3$ by $4$ .

$f (4) = - 12 + \frac{21}{2}$

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

$f (4) = - 12 \cdot \frac{2}{2} + \frac{21}{2}$

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

$f (4) = \frac{- 12 \cdot 2}{2} + \frac{21}{2}$

Combine the numerators over the common denominator.

$f (4) = \frac{- 12 \cdot 2 + 21}{2}$

Simplify the numerator.

Multiply $- 12$ by $2$ .

$f (4) = \frac{- 24 + 21}{2}$

Add $- 24$ and $21$ .

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

Move the negative in front of the fraction.

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

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

$- \frac{3}{2}$

The $y$ value at $x = 4$ is $- \frac{3}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & - \frac{3}{2} \\ 3 & - 2 \\ 4 & - \frac{3}{2} \\ 5 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(3, - 2)$

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

Axis of Symmetry: $x = 3$

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

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & - \frac{3}{2} \\ 3 & - 2 \\ 4 & - \frac{3}{2} \\ 5 & 0 \end{array}$

Step 4

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

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