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

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

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

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

Apply the distributive property.

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

Simplify.

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 4 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

$b = - 2$

$c = 2$

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

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

Simplify the right side.

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 (\frac{1}{2})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 (\frac{1}{2})}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

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

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

Simplify the right side.

Simplify each term.

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

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

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

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

Divide $4$ by $2$ .

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

Divide $4$ by $2$ .

$e = 2 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = 2 - 2$

Subtract $2$ from $2$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 2)}^{2} + 0$ .

$\frac{1}{2} {(x - 2)}^{2} + 0$

Set $y$ equal to the new right side.

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

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

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

$h = 2$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(2, 0)$

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.

$(2, \frac{1}{2})$

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.

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

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

Direction: Opens Up

Vertex: $(2, 0)$

Focus: $(2, \frac{1}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{1}{2}$

Direction: Opens Up

Vertex: $(2, 0)$

Focus: $(2, \frac{1}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{1}{2}$

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.

$f (0) = \frac{{(0)}^{2}}{2} - 2 \cdot 0 + 2$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{0}{2} - 2 \cdot 0 + 2$

Divide $0$ by $2$ .

$f (0) = 0 - 2 \cdot 0 + 2$

Multiply $- 2$ by $0$ .

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

Simplify by adding numbers.

Add $0$ and $0$ .

$f (0) = 0 + 2$

Add $0$ and $2$ .

$f (0) = 2$

The final answer is $2$ .

$2$

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

$y = 2$

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

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

Simplify the result.

Simplify each term.

One to any power is one.

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

Multiply $- 2$ by $1$ .

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

Find the common denominator.

Write $- 2$ as a fraction with denominator $1$ .

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

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

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

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

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

Write $2$ as a fraction with denominator $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 2$ by $2$ .

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

Multiply $2$ by $2$ .

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

Simplify by adding and subtracting.

Subtract $4$ from $1$ .

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

Add $- 3$ and $4$ .

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

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

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

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

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

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

Simplify the result.

Simplify each term.

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

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

Divide $16$ by $2$ .

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

Multiply $- 2$ by $4$ .

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

Simplify by adding and subtracting.

Subtract $8$ from $8$ .

$f (4) = 0 + 2$

Add $0$ and $2$ .

$f (4) = 2$

The final answer is $2$ .

$2$

The $y$ value at $x = 4$ is $2$ .

$y = 2$

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

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

Simplify the result.

Simplify each term.

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

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

Multiply $- 2$ by $3$ .

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

Find the common denominator.

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

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

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

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

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

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

Write $2$ as a fraction with denominator $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 6$ by $2$ .

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

Multiply $2$ by $2$ .

$f (3) = \frac{9 - 12 + 4}{2}$

Simplify by adding and subtracting.

Subtract $12$ from $9$ .

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

Add $- 3$ and $4$ .

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

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

The $y$ value at $x = 3$ is $\frac{1}{2}$ .

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

Graph the parabola using its properties and the selected points.

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

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(2, 0)$

Focus: $(2, \frac{1}{2})$

Axis of Symmetry: $x = 2$

Directrix: $y = - \frac{1}{2}$

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

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

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