Graph f(x)=-4x^2-40x-96

$f (x) = - 4 x^{2} - 40 x - 96$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 4 x^{2} - 40 x - 96$ .

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

$a = - 4$

$b = - 40$

$c = - 96$

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

$d = \frac{- 40}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $- 40$ and $2$ .

Factor $2$ out of $- 40$ .

$d = \frac{2 \cdot - 20}{2 \cdot - 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 4$ .

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

Cancel the common factor.

$d = \frac{2 \cdot - 20}{2 \cdot - 4}$

Rewrite the expression.

$d = \frac{- 20}{- 4}$

Cancel the common factor of $- 20$ and $- 4$ .

Factor $- 4$ out of $- 20$ .

$d = \frac{- 4 (5)}{- 4}$

Cancel the common factors.

Factor $- 4$ out of $- 4$ .

$d = \frac{- 4 \cdot 5}{- 4 \cdot 1}$

Cancel the common factor.

$d = \frac{- 4 \cdot 5}{- 4 \cdot 1}$

Rewrite the expression.

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

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

$e = - 96 - \frac{{(- 40)}^{2}}{4 \cdot - 4}$

Simplify the right side.

Simplify each term.

Raise $- 40$ to the power of $2$ .

$e = - 96 - \frac{1600}{4 \cdot - 4}$

Multiply $4$ by $- 4$ .

$e = - 96 - \frac{1600}{- 16}$

Divide $1600$ by $- 16$ .

$e = - 96 - - 100$

Multiply $- 1$ by $- 100$ .

$e = - 96 + 100$

Add $- 96$ and $100$ .

$e = 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + 5)}^{2} + 4$ .

$- 4 {(x + 5)}^{2} + 4$

Set $y$ equal to the new right side.

$y = - 4 {(x + 5)}^{2} + 4$

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

$a = - 4$

$h = - 5$

$k = 4$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 5, 4)$

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.

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

Simplify.

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Move the negative in front of the fraction.

$- \frac{1}{16}$

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.

$(- 5, \frac{63}{16})$

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.

$y = \frac{65}{16}$

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

Direction: Opens Down

Vertex: $(- 5, 4)$

Focus: $(- 5, \frac{63}{16})$

Axis of Symmetry: $x = - 5$

Directrix: $y = \frac{65}{16}$

Direction: Opens Down

Vertex: $(- 5, 4)$

Focus: $(- 5, \frac{63}{16})$

Axis of Symmetry: $x = - 5$

Directrix: $y = \frac{65}{16}$

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 $- 6$ in the expression.

$f (- 6) = - 4 {(- 6)}^{2} - 40 \cdot - 6 - 96$

Simplify the result.

Simplify each term.

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

$f (- 6) = - 4 \cdot 36 - 40 \cdot - 6 - 96$

Multiply $- 4$ by $36$ .

$f (- 6) = - 144 - 40 \cdot - 6 - 96$

Multiply $- 40$ by $- 6$ .

$f (- 6) = - 144 + 240 - 96$

Simplify by adding and subtracting.

Add $- 144$ and $240$ .

$f (- 6) = 96 - 96$

Subtract $96$ from $96$ .

$f (- 6) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 6$ is $0$ .

$y = 0$

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

$f (- 7) = - 4 {(- 7)}^{2} - 40 \cdot - 7 - 96$

Simplify the result.

Simplify each term.

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

$f (- 7) = - 4 \cdot 49 - 40 \cdot - 7 - 96$

Multiply $- 4$ by $49$ .

$f (- 7) = - 196 - 40 \cdot - 7 - 96$

Multiply $- 40$ by $- 7$ .

$f (- 7) = - 196 + 280 - 96$

Simplify by adding and subtracting.

Add $- 196$ and $280$ .

$f (- 7) = 84 - 96$

Subtract $96$ from $84$ .

$f (- 7) = - 12$

The final answer is $- 12$ .

$- 12$

The $y$ value at $x = - 7$ is $- 12$ .

$y = - 12$

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

$f (- 4) = - 4 {(- 4)}^{2} - 40 \cdot - 4 - 96$

Simplify the result.

Simplify each term.

Multiply $- 4$ by ${(- 4)}^{2}$ by adding the exponents.

Multiply $- 4$ by ${(- 4)}^{2}$ .

Raise $- 4$ to the power of $1$ .

$f (- 4) = (- 4) {(- 4)}^{2} - 40 \cdot - 4 - 96$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 4) = {(- 4)}^{1 + 2} - 40 \cdot - 4 - 96$

Add $1$ and $2$ .

$f (- 4) = {(- 4)}^{3} - 40 \cdot - 4 - 96$

Raise $- 4$ to the power of $3$ .

$f (- 4) = - 64 - 40 \cdot - 4 - 96$

Multiply $- 40$ by $- 4$ .

$f (- 4) = - 64 + 160 - 96$

Simplify by adding and subtracting.

Add $- 64$ and $160$ .

$f (- 4) = 96 - 96$

Subtract $96$ from $96$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 4 {(- 3)}^{2} - 40 \cdot - 3 - 96$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 4 \cdot 9 - 40 \cdot - 3 - 96$

Multiply $- 4$ by $9$ .

$f (- 3) = - 36 - 40 \cdot - 3 - 96$

Multiply $- 40$ by $- 3$ .

$f (- 3) = - 36 + 120 - 96$

Simplify by adding and subtracting.

Add $- 36$ and $120$ .

$f (- 3) = 84 - 96$

Subtract $96$ from $84$ .

$f (- 3) = - 12$

The final answer is $- 12$ .

$- 12$

The $y$ value at $x = - 3$ is $- 12$ .

$y = - 12$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 7 & - 12 \\ - 6 & 0 \\ - 5 & 4 \\ - 4 & 0 \\ - 3 & - 12 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 5, 4)$

Focus: $(- 5, \frac{63}{16})$

Axis of Symmetry: $x = - 5$

Directrix: $y = \frac{65}{16}$

$\begin{array}{c} x & y \\ - 7 & - 12 \\ - 6 & 0 \\ - 5 & 4 \\ - 4 & 0 \\ - 3 & - 12 \end{array}$

Step 4

Graph f(x)=-4x^2-40x-96

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