Graph y=-(1/2x)^2+5

$y = - {(\frac{1}{2} x)}^{2} + 5$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Simplify each term.

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

$y = - {(\frac{x}{2})}^{2} + 5$

Apply the product rule to $\frac{x}{2}$ .

$y = - \frac{x^{2}}{2^{2}} + 5$

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

$y = - \frac{x^{2}}{4} + 5$

Complete the square for $- \frac{x^{2}}{4} + 5$ .

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

$a = - \frac{1}{4}$

$b = 0$

$c = 5$

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

$d = \frac{0}{2 (- \frac{1}{4})}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{4})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{4})}$

Rewrite the expression.

$d = \frac{0}{- \frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 4)$

Multiply $0 (- 1 \cdot 4)$ .

Multiply $- 1$ by $4$ .

$d = 0 \cdot - 4$

Multiply $0$ by $- 4$ .

$d = 0$

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

$e = 5 - \frac{0^{2}}{4 (- \frac{1}{4})}$

Simplify the right side.

Simplify each term.

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

$e = 5 - \frac{0}{4 (- \frac{1}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 5 - \frac{0}{- 4 (\frac{1}{4})}$

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

$e = 5 - \frac{0}{\frac{- 4}{4}}$

Divide $- 4$ by $4$ .

$e = 5 - \frac{0}{- 1}$

Divide $0$ by $- 1$ .

$e = 5 - 0$

Multiply $- 1$ by $0$ .

$e = 5 + 0$

Add $5$ and $0$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 0)}^{2} + 5$ .

$- \frac{1}{4} {(x + 0)}^{2} + 5$

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 0)}^{2} + 5$

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

$a = - \frac{1}{4}$

$h = 0$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(0, 5)$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

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

$- \frac{1}{\frac{4}{4}}$

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$

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.

$(0, 4)$

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

$x = 0$

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.

$y = 6$

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

Direction: Opens Down

Vertex: $(0, 5)$

Focus: $(0, 4)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

Direction: Opens Down

Vertex: $(0, 5)$

Focus: $(0, 4)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

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 $- 2$ in the expression.

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

Simplify the result.

Simplify each term.

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$f (- 2) = - 1 \cdot 1 + 5$

Multiply $- 1$ by $1$ .

$f (- 2) = - 1 + 5$

Add $- 1$ and $5$ .

$f (- 2) = 4$

The final answer is $4$ .

$4$

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

$y = 4$

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

$f (- 1) = - \frac{{(- 1)}^{2}}{4} + 5$

Simplify the result.

Raise $- 1$ to the power of $2$ .

$f (- 1) = - \frac{1}{4} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 1) = - \frac{1}{4} + 5 \cdot \frac{4}{4}$

Combine $5$ and $\frac{4}{4}$ .

$f (- 1) = - \frac{1}{4} + \frac{5 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 1 + 5 \cdot 4}{4}$

Simplify the numerator.

Multiply $5$ by $4$ .

$f (- 1) = \frac{- 1 + 20}{4}$

Add $- 1$ and $20$ .

$f (- 1) = \frac{19}{4}$

The final answer is $\frac{19}{4}$ .

$\frac{19}{4}$

The $y$ value at $x = - 1$ is $\frac{19}{4}$ .

$y = \frac{19}{4}$

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

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

Simplify the result.

Simplify each term.

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$f (2) = - 1 \cdot 1 + 5$

Multiply $- 1$ by $1$ .

$f (2) = - 1 + 5$

Add $- 1$ and $5$ .

$f (2) = 4$

The final answer is $4$ .

$4$

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

$y = 4$

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

$f (1) = - \frac{{(1)}^{2}}{4} + 5$

Simplify the result.

One to any power is one.

$f (1) = - \frac{1}{4} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (1) = - \frac{1}{4} + 5 \cdot \frac{4}{4}$

Combine $5$ and $\frac{4}{4}$ .

$f (1) = - \frac{1}{4} + \frac{5 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (1) = \frac{- 1 + 5 \cdot 4}{4}$

Simplify the numerator.

Multiply $5$ by $4$ .

$f (1) = \frac{- 1 + 20}{4}$

Add $- 1$ and $20$ .

$f (1) = \frac{19}{4}$

The final answer is $\frac{19}{4}$ .

$\frac{19}{4}$

The $y$ value at $x = 1$ is $\frac{19}{4}$ .

$y = \frac{19}{4}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 4 \\ - 1 & \frac{19}{4} \\ 0 & 5 \\ 1 & \frac{19}{4} \\ 2 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(0, 5)$

Focus: $(0, 4)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

$\begin{array}{c} x & y \\ - 2 & 4 \\ - 1 & \frac{19}{4} \\ 0 & 5 \\ 1 & \frac{19}{4} \\ 2 & 4 \end{array}$

Step 4

Graph y=-(1/2x)^2+5

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