Graph y=-0.0005x^2+1.2345x+-2.2

$y = - 0.0005 x^{2} + 1.2345 x + - 2.2$

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

$y = - 0.0005 x^{2} + 1.2345 x - 2.2$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 0.0005 x^{2} + 1.2345 x - 2.2$ .

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

$a = - 0.0005$

$b = 1.2345$

$c = - 2.2$

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

$d = \frac{1.2345}{2 \cdot - 0.0005}$

Simplify the right side.

Multiply $2$ by $- 0.0005$ .

$d = \frac{1.2345}{- 0.001}$

Divide $1.2345$ by $- 0.001$ .

$d = - 1234.5$

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

$e = - 2.2 - \frac{{1.2345}^{2}}{4 \cdot - 0.0005}$

Simplify the right side.

Simplify each term.

Raise $1.2345$ to the power of $2$ .

$e = - 2.2 - \frac{1.52399025}{4 \cdot - 0.0005}$

Multiply $4$ by $- 0.0005$ .

$e = - 2.2 - \frac{1.52399025}{- 0.002}$

Divide $1.52399025$ by $- 0.002$ .

$e = - 2.2 - - 761.995125$

Multiply $- 1$ by $- 761.995125$ .

$e = - 2.2 + 761.995125$

Add $- 2.2$ and $761.995125$ .

$e = 759.795125$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 0.0005 {(x - 1234.5)}^{2} + 759.795125$ .

$- 0.0005 {(x - 1234.5)}^{2} + 759.795125$

Set $y$ equal to the new right side.

$y = - 0.0005 {(x - 1234.5)}^{2} + 759.795125$

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

$a = - 0.0005$

$h = 1234.5$

$k = 759.795125$

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

Opens Down

Find the vertex $(h, k)$ .

$(1234.5, 759.795125)$

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.

$\frac{1}{4 \cdot - 0.0005}$

Simplify.

Multiply $4$ by $- 0.0005$ .

$\frac{1}{- 0.002}$

Divide $1$ by $- 0.002$ .

$- 500$

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.

$(1234.5, 259.795125)$

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

$x = 1234.5$

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.

$y = 1259.795125$

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

Direction: Opens Down

Vertex: $(1234.5, 759.795125)$

Focus: $(1234.5, 259.795125)$

Axis of Symmetry: $x = 1234.5$

Directrix: $y = 1259.795125$

Direction: Opens Down

Vertex: $(1234.5, 759.795125)$

Focus: $(1234.5, 259.795125)$

Axis of Symmetry: $x = 1234.5$

Directrix: $y = 1259.795125$

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 $1233$ in the expression.

$f (1233) = - 0.0005 {(1233)}^{2} + 1.2345 (1233) - 2.2$

Simplify the result.

Simplify each term.

Raise $1233$ to the power of $2$ .

$f (1233) = - 0.0005 \cdot 1520289 + 1.2345 (1233) - 2.2$

Multiply $- 0.0005$ by $1520289$ .

$f (1233) = - 760.1445 + 1.2345 (1233) - 2.2$

Multiply $1.2345$ by $1233$ .

$f (1233) = - 760.1445 + 1522.1385 - 2.2$

Simplify by adding and subtracting.

Add $- 760.1445$ and $1522.1385$ .

$f (1233) = 761.994 - 2.2$

Subtract $2.2$ from $761.994$ .

$f (1233) = 759.794$

The final answer is $759.794$ .

$759.794$

The $y$ value at $x = 1233$ is $759.794$ .

$y = 759.794$

Replace the variable $x$ with $1232$ in the expression.

$f (1232) = - 0.0005 {(1232)}^{2} + 1.2345 (1232) - 2.2$

Simplify the result.

Simplify each term.

Raise $1232$ to the power of $2$ .

$f (1232) = - 0.0005 \cdot 1517824 + 1.2345 (1232) - 2.2$

Multiply $- 0.0005$ by $1517824$ .

$f (1232) = - 758.912 + 1.2345 (1232) - 2.2$

Multiply $1.2345$ by $1232$ .

$f (1232) = - 758.912 + 1520.904 - 2.2$

Simplify by adding and subtracting.

Add $- 758.912$ and $1520.904$ .

$f (1232) = 761.992 - 2.2$

Subtract $2.2$ from $761.992$ .

$f (1232) = 759.792$

The final answer is $759.792$ .

$759.792$

The $y$ value at $x = 1232$ is $759.792$ .

$y = 759.792$

Replace the variable $x$ with $1235$ in the expression.

$f (1235) = - 0.0005 {(1235)}^{2} + 1.2345 (1235) - 2.2$

Simplify the result.

Simplify each term.

Raise $1235$ to the power of $2$ .

$f (1235) = - 0.0005 \cdot 1525225 + 1.2345 (1235) - 2.2$

Multiply $- 0.0005$ by $1525225$ .

$f (1235) = - 762.6125 + 1.2345 (1235) - 2.2$

Multiply $1.2345$ by $1235$ .

$f (1235) = - 762.6125 + 1524.6075 - 2.2$

Simplify by adding and subtracting.

Add $- 762.6125$ and $1524.6075$ .

$f (1235) = 761.995 - 2.2$

Subtract $2.2$ from $761.995$ .

$f (1235) = 759.795$

The final answer is $759.795$ .

$759.795$

The $y$ value at $x = 1235$ is $759.795$ .

$y = 759.795$

Replace the variable $x$ with $1236$ in the expression.

$f (1236) = - 0.0005 {(1236)}^{2} + 1.2345 (1236) - 2.2$

Simplify the result.

Simplify each term.

Raise $1236$ to the power of $2$ .

$f (1236) = - 0.0005 \cdot 1527696 + 1.2345 (1236) - 2.2$

Multiply $- 0.0005$ by $1527696$ .

$f (1236) = - 763.848 + 1.2345 (1236) - 2.2$

Multiply $1.2345$ by $1236$ .

$f (1236) = - 763.848 + 1525.842 - 2.2$

Simplify by adding and subtracting.

Add $- 763.848$ and $1525.842$ .

$f (1236) = 761.994 - 2.2$

Subtract $2.2$ from $761.994$ .

$f (1236) = 759.794$

The final answer is $759.794$ .

$759.794$

The $y$ value at $x = 1236$ is $759.794$ .

$y = 759.794$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1232 & 759.792 \\ 1233 & 759.794 \\ 1234.5 & 759.795 \\ 1235 & 759.795 \\ 1236 & 759.794 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1234.5, 759.795125)$

Focus: $(1234.5, 259.795125)$

Axis of Symmetry: $x = 1234.5$

Directrix: $y = 1259.795125$

$\begin{array}{c} x & y \\ 1232 & 759.792 \\ 1233 & 759.794 \\ 1234.5 & 759.795 \\ 1235 & 759.795 \\ 1236 & 759.794 \end{array}$

Graph y=-0.0005x^2+1.2345x+-2.2

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