Graph y=-1/2(x-3)(x-7)

$y = - \frac{1}{2} (x - 3) (x - 7)$

Solve for $y$ .

Remove parentheses.

$y = - \frac{1}{2} \cdot ((x - 3) (x - 7))$

Remove parentheses.

$y = - \frac{1}{2} \cdot ((x - 3) (x - 7))$

Simplify $- \frac{1}{2} \cdot ((x - 3) (x - 7))$ .

Expand $(x - 3) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$y = - \frac{1}{2} \cdot (x (x - 7) - 3 (x - 7))$

Apply the distributive property.

$y = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 7 - 3 (x - 7))$

Apply the distributive property.

$y = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 7 - 3 x - 3 \cdot - 7)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - \frac{1}{2} \cdot (x^{2} + x \cdot - 7 - 3 x - 3 \cdot - 7)$

Move $- 7$ to the left of $x$ .

$y = - \frac{1}{2} \cdot (x^{2} - 7 \cdot x - 3 x - 3 \cdot - 7)$

Multiply $- 3$ by $- 7$ .

$y = - \frac{1}{2} \cdot (x^{2} - 7 x - 3 x + 21)$

Subtract $3 x$ from $- 7 x$ .

$y = - \frac{1}{2} \cdot (x^{2} - 10 x + 21)$

Apply the distributive property.

$y = - \frac{1}{2} x^{2} - \frac{1}{2} (- 10 x) - \frac{1}{2} \cdot 21$

Simplify.

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

$y = - \frac{x^{2}}{2} - \frac{1}{2} (- 10 x) - \frac{1}{2} \cdot 21$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y = - \frac{x^{2}}{2} + \frac{- 1}{2} (- 10 x) - \frac{1}{2} \cdot 21$

Factor $2$ out of $- 10 x$ .

$y = - \frac{x^{2}}{2} + \frac{- 1}{2} (2 (- 5 x)) - \frac{1}{2} \cdot 21$

Cancel the common factor.

$y = - \frac{x^{2}}{2} + \frac{- 1}{2} (2 (- 5 x)) - \frac{1}{2} \cdot 21$

Rewrite the expression.

$y = - \frac{x^{2}}{2} - 1 (- 5 x) - \frac{1}{2} \cdot 21$

Multiply $- 5$ by $- 1$ .

$y = - \frac{x^{2}}{2} + 5 x - \frac{1}{2} \cdot 21$

Multiply $- \frac{1}{2} \cdot 21$ .

Multiply $21$ by $- 1$ .

$y = - \frac{x^{2}}{2} + 5 x - 21 (\frac{1}{2})$

Combine $- 21$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} + 5 x + \frac{- 21}{2}$

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{2} + 5 x - \frac{21}{2}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{x^{2}}{2} + 5 x - \frac{21}{2}$ .

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

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

$b = 5$

$c = - \frac{21}{2}$

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

$d = \frac{5}{2 (- \frac{1}{2})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{5}{- 2 (\frac{1}{2})}$

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

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

Divide $- 2$ by $2$ .

$d = \frac{5}{- 1}$

Divide $5$ by $- 1$ .

$d = - 5$

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

$e = - \frac{21}{2} - \frac{5^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = - \frac{21}{2} - \frac{25}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{21}{2} - \frac{25}{- 4 (\frac{1}{2})}$

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

$e = - \frac{21}{2} - \frac{25}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - \frac{21}{2} - \frac{25}{- 2}$

Move the negative in front of the fraction.

$e = - \frac{21}{2} - - \frac{25}{2}$

Multiply $- - \frac{25}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{21}{2} + 1 (\frac{25}{2})$

Multiply $\frac{25}{2}$ by $1$ .

$e = - \frac{21}{2} + \frac{25}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 21 + 25}{2}$

Add $- 21$ and $25$ .

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

Divide $4$ by $2$ .

$e = 2$

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

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

Set $y$ equal to the new right side.

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

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

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

$h = 5$

$k = 2$

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

Opens Down

Find the vertex $(h, k)$ .

$(5, 2)$

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.

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.

$(5, \frac{3}{2})$

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

$x = 5$

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 Down

Vertex: $(5, 2)$

Focus: $(5, \frac{3}{2})$

Axis of Symmetry: $x = 5$

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

Direction: Opens Down

Vertex: $(5, 2)$

Focus: $(5, \frac{3}{2})$

Axis of Symmetry: $x = 5$

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 $3$ in the expression.

$f (3) = - \frac{{(3)}^{2}}{2} + 5 (3) - \frac{21}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (3) = 5 (3) + \frac{- {(3)}^{2} - 21}{2}$

Simplify each term.

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

$f (3) = 5 (3) + \frac{- 1 \cdot 9 - 21}{2}$

Multiply $- 1$ by $9$ .

$f (3) = 5 (3) + \frac{- 9 - 21}{2}$

Subtract $21$ from $- 9$ .

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

Simplify each term.

Multiply $5$ by $3$ .

$f (3) = 15 + \frac{- 30}{2}$

Divide $- 30$ by $2$ .

$f (3) = 15 - 15$

Subtract $15$ from $15$ .

$f (3) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 3$ is $0$ .

$y = 0$

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

$f (4) = - \frac{{(4)}^{2}}{2} + 5 (4) - \frac{21}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (4) = 5 (4) + \frac{- {(4)}^{2} - 21}{2}$

Simplify each term.

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

$f (4) = 5 (4) + \frac{- 1 \cdot 16 - 21}{2}$

Multiply $- 1$ by $16$ .

$f (4) = 5 (4) + \frac{- 16 - 21}{2}$

Subtract $21$ from $- 16$ .

$f (4) = 5 (4) + \frac{- 37}{2}$

Simplify each term.

Multiply $5$ by $4$ .

$f (4) = 20 + \frac{- 37}{2}$

Move the negative in front of the fraction.

$f (4) = 20 - \frac{37}{2}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (4) = 20 \cdot \frac{2}{2} - \frac{37}{2}$

Combine $20$ and $\frac{2}{2}$ .

$f (4) = \frac{20 \cdot 2}{2} - \frac{37}{2}$

Combine the numerators over the common denominator.

$f (4) = \frac{20 \cdot 2 - 37}{2}$

Simplify the numerator.

Multiply $20$ by $2$ .

$f (4) = \frac{40 - 37}{2}$

Subtract $37$ from $40$ .

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

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

$\frac{3}{2}$

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

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

Replace the variable $x$ with $7$ in the expression.

$f (7) = - \frac{{(7)}^{2}}{2} + 5 (7) - \frac{21}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (7) = 5 (7) + \frac{- {(7)}^{2} - 21}{2}$

Simplify each term.

Raise $7$ to the power of $2$ .

$f (7) = 5 (7) + \frac{- 1 \cdot 49 - 21}{2}$

Multiply $- 1$ by $49$ .

$f (7) = 5 (7) + \frac{- 49 - 21}{2}$

Subtract $21$ from $- 49$ .

$f (7) = 5 (7) + \frac{- 70}{2}$

Simplify each term.

Multiply $5$ by $7$ .

$f (7) = 35 + \frac{- 70}{2}$

Divide $- 70$ by $2$ .

$f (7) = 35 - 35$

Subtract $35$ from $35$ .

$f (7) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 7$ is $0$ .

$y = 0$

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

$f (6) = - \frac{{(6)}^{2}}{2} + 5 (6) - \frac{21}{2}$

Simplify the result.

Combine the numerators over the common denominator.

$f (6) = 5 (6) + \frac{- {(6)}^{2} - 21}{2}$

Simplify each term.

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

$f (6) = 5 (6) + \frac{- 1 \cdot 36 - 21}{2}$

Multiply $- 1$ by $36$ .

$f (6) = 5 (6) + \frac{- 36 - 21}{2}$

Subtract $21$ from $- 36$ .

$f (6) = 5 (6) + \frac{- 57}{2}$

Simplify each term.

Multiply $5$ by $6$ .

$f (6) = 30 + \frac{- 57}{2}$

Move the negative in front of the fraction.

$f (6) = 30 - \frac{57}{2}$

To write $30$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (6) = 30 \cdot \frac{2}{2} - \frac{57}{2}$

Combine $30$ and $\frac{2}{2}$ .

$f (6) = \frac{30 \cdot 2}{2} - \frac{57}{2}$

Combine the numerators over the common denominator.

$f (6) = \frac{30 \cdot 2 - 57}{2}$

Simplify the numerator.

Multiply $30$ by $2$ .

$f (6) = \frac{60 - 57}{2}$

Subtract $57$ from $60$ .

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

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

$\frac{3}{2}$

The $y$ value at $x = 6$ is $\frac{3}{2}$ .

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

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & 0 \\ 4 & \frac{3}{2} \\ 5 & 2 \\ 6 & \frac{3}{2} \\ 7 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(5, 2)$

Focus: $(5, \frac{3}{2})$

Axis of Symmetry: $x = 5$

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

$\begin{array}{c} x & y \\ 3 & 0 \\ 4 & \frac{3}{2} \\ 5 & 2 \\ 6 & \frac{3}{2} \\ 7 & 0 \end{array}$

Graph y=-1/2(x-3)(x-7)

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