Graph P=-25x^2+300x

$P = - 25 x^{2} + 300 x$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 25 x^{2} + 300 x$ .

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

$a = - 25$

$b = 300$

$c = 0$

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

$d = \frac{300}{2 \cdot - 25}$

Simplify the right side.

Cancel the common factor of $300$ and $2$ .

Factor $2$ out of $300$ .

$d = \frac{2 \cdot 150}{2 \cdot - 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 25$ .

$d = \frac{2 \cdot 150}{2 (- 25)}$

Cancel the common factor.

$d = \frac{2 \cdot 150}{2 \cdot - 25}$

Rewrite the expression.

$d = \frac{150}{- 25}$

Cancel the common factor of $150$ and $- 25$ .

Factor $25$ out of $150$ .

$d = \frac{25 (6)}{- 25}$

Move the negative one from the denominator of $\frac{6}{- 1}$ .

$d = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$d = - 6$

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

$e = 0 - \frac{300^{2}}{4 \cdot - 25}$

Simplify the right side.

Simplify each term.

Raise $300$ to the power of $2$ .

$e = 0 - \frac{90000}{4 \cdot - 25}$

Multiply $4$ by $- 25$ .

$e = 0 - \frac{90000}{- 100}$

Divide $90000$ by $- 100$ .

$e = 0 - - 900$

Multiply $- 1$ by $- 900$ .

$e = 0 + 900$

Add $0$ and $900$ .

$e = 900$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(x - 6)}^{2} + 900$ .

$- 25 {(x - 6)}^{2} + 900$

Set $y$ equal to the new right side.

$y = - 25 {(x - 6)}^{2} + 900$

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

$a = - 25$

$h = 6$

$k = 900$

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

Opens Down

Find the vertex $(h, k)$ .

$(6, 900)$

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.

$\frac{1}{4 \cdot - 25}$

Simplify.

Multiply $4$ by $- 25$ .

$\frac{1}{- 100}$

Move the negative in front of the fraction.

$- \frac{1}{100}$

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.

$(6, \frac{89999}{100})$

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

$x = 6$

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.

$y = \frac{90001}{100}$

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

Direction: Opens Down

Vertex: $(6, 900)$

Focus: $(6, \frac{89999}{100})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{90001}{100}$

Direction: Opens Down

Vertex: $(6, 900)$

Focus: $(6, \frac{89999}{100})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{90001}{100}$

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 $5$ in the expression.

$f (5) = - 25 {(5)}^{2} + 300 (5)$

Simplify the result.

Simplify each term.

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

$f (5) = - 25 \cdot 25 + 300 (5)$

Multiply $- 25$ by $25$ .

$f (5) = - 625 + 300 (5)$

Multiply $300$ by $5$ .

$f (5) = - 625 + 1500$

Add $- 625$ and $1500$ .

$f (5) = 875$

The final answer is $875$ .

$875$

The $y$ value at $x = 5$ is $875$ .

$y = 875$

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

$f (7) = - 25 {(7)}^{2} + 300 (7)$

Simplify the result.

Simplify each term.

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

$f (7) = - 25 \cdot 49 + 300 (7)$

Multiply $- 25$ by $49$ .

$f (7) = - 1225 + 300 (7)$

Multiply $300$ by $7$ .

$f (7) = - 1225 + 2100$

Add $- 1225$ and $2100$ .

$f (7) = 875$

The final answer is $875$ .

$875$

The $y$ value at $x = 7$ is $875$ .

$y = 875$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 5 & 875 \\ 6 & 900 \\ 7 & 875 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(6, 900)$

Focus: $(6, \frac{89999}{100})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{90001}{100}$

$\begin{array}{c} x & y \\ 5 & 875 \\ 6 & 900 \\ 7 & 875 \end{array}$

Step 4

Graph P=-25x^2+300x

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