Graph y=-1/2(x-3)^2+5

$y = - \frac{1}{2} {(x - 3)}^{2} + 5$

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 $k$ .

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

$h = 3$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(3, 5)$

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.

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

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.

$(3, \frac{9}{2})$

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.

$y = \frac{11}{2}$

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

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

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.

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

One to any power is one.

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

Multiply $- 1$ by $1$ .

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

Add $- 1$ and $1$ .

$f (1) = 3 (1) + \frac{0}{2}$

Simplify each term.

Multiply $3$ by $1$ .

$f (1) = 3 + \frac{0}{2}$

Divide $0$ by $2$ .

$f (1) = 3 + 0$

Add $3$ and $0$ .

$f (1) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 1$ is $3$ .

$y = 3$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Multiply $- 1$ by $4$ .

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

Add $- 4$ and $1$ .

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

Simplify each term.

Multiply $3$ by $2$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $6$ by $2$ .

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

Subtract $3$ from $12$ .

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

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

$\frac{9}{2}$

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

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

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (5) = 3 (5) + \frac{- 1 \cdot 25 + 1}{2}$

Multiply $- 1$ by $25$ .

$f (5) = 3 (5) + \frac{- 25 + 1}{2}$

Add $- 25$ and $1$ .

$f (5) = 3 (5) + \frac{- 24}{2}$

Simplify each term.

Multiply $3$ by $5$ .

$f (5) = 15 + \frac{- 24}{2}$

Divide $- 24$ by $2$ .

$f (5) = 15 - 12$

Subtract $12$ from $15$ .

$f (5) = 3$

The final answer is $3$ .

$3$

The $y$ value at $x = 5$ is $3$ .

$y = 3$

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Simplify each term.

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

$f (4) = 3 (4) + \frac{- 1 \cdot 16 + 1}{2}$

Multiply $- 1$ by $16$ .

$f (4) = 3 (4) + \frac{- 16 + 1}{2}$

Add $- 16$ and $1$ .

$f (4) = 3 (4) + \frac{- 15}{2}$

Simplify each term.

Multiply $3$ by $4$ .

$f (4) = 12 + \frac{- 15}{2}$

Move the negative in front of the fraction.

$f (4) = 12 - \frac{15}{2}$

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

$f (4) = 12 \cdot \frac{2}{2} - \frac{15}{2}$

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

$f (4) = \frac{12 \cdot 2}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$f (4) = \frac{12 \cdot 2 - 15}{2}$

Simplify the numerator.

Multiply $12$ by $2$ .

$f (4) = \frac{24 - 15}{2}$

Subtract $15$ from $24$ .

$f (4) = \frac{9}{2}$

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

$\frac{9}{2}$

The $y$ value at $x = 4$ is $\frac{9}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 3 \\ 2 & \frac{9}{2} \\ 3 & 5 \\ 4 & \frac{9}{2} \\ 5 & 3 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

$\begin{array}{c} x & y \\ 1 & 3 \\ 2 & \frac{9}{2} \\ 3 & 5 \\ 4 & \frac{9}{2} \\ 5 & 3 \end{array}$

Step 4

Graph y=-1/2(x-3)^2+5

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