Find the Properties h(x)=-3x^2

$h (x) = - 3 x^{2}$

Step 1

Write $h (x) = - 3 x^{2}$ as an equation.

$y = - 3 x^{2}$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 3 x^{2}$ .

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

$a = - 3$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 \cdot - 3}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 (0)}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{0}{- 3}$

Cancel the common factor of $0$ and $- 3$ .

Factor $3$ out of $0$ .

$d = \frac{3 (0)}{- 3}$

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Multiply $- 1$ by $0$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = 0 - \frac{0}{- 12}$

Divide $0$ by $- 12$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 x^{2}$ .

$- 3 x^{2}$

Set $y$ equal to the new right side.

$y = - 3 x^{2}$

Step 3

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

$a = - 3$

$h = 0$

$k = 0$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(0, 0)$

Step 6

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.

$\frac{1}{4 \cdot - 3}$

Simplify.

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 7

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.

$(0, - \frac{1}{12})$

Step 8

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

$x = 0$

Step 9

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.

$y = \frac{1}{12}$

Step 10

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

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{12})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{1}{12}$

Step 11

Find the Properties h(x)=-3x^2

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