Graph y=-5x^2-10x-15

$y = - 5 x^{2} - 10 x - 15$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 5 x^{2} - 10 x - 15$ .

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

$a = - 5$

$b = - 10$

$c = - 15$

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

$d = \frac{- 10}{2 \cdot - 5}$

Simplify the right side.

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 5$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = - 15 - \frac{{(- 10)}^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = - 15 - \frac{100}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = - 15 - \frac{100}{- 20}$

Divide $100$ by $- 20$ .

$e = - 15 - - 5$

Multiply $- 1$ by $- 5$ .

$e = - 15 + 5$

Add $- 15$ and $5$ .

$e = - 10$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x + 1)}^{2} - 10$ .

$- 5 {(x + 1)}^{2} - 10$

Set $y$ equal to the new right side.

$y = - 5 {(x + 1)}^{2} - 10$

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

$a = - 5$

$h = - 1$

$k = - 10$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 1, - 10)$

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.

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

Simplify.

Multiply $4$ by $- 5$ .

$\frac{1}{- 20}$

Move the negative in front of the fraction.

$- \frac{1}{20}$

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.

$(- 1, - \frac{201}{20})$

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

$x = - 1$

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.

$y = - \frac{199}{20}$

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

Direction: Opens Down

Vertex: $(- 1, - 10)$

Focus: $(- 1, - \frac{201}{20})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{199}{20}$

Direction: Opens Down

Vertex: $(- 1, - 10)$

Focus: $(- 1, - \frac{201}{20})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{199}{20}$

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 $- 2$ in the expression.

$f (- 2) = - 5 {(- 2)}^{2} - 10 \cdot - 2 - 15$

Simplify the result.

Simplify each term.

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

$f (- 2) = - 5 \cdot 4 - 10 \cdot - 2 - 15$

Multiply $- 5$ by $4$ .

$f (- 2) = - 20 - 10 \cdot - 2 - 15$

Multiply $- 10$ by $- 2$ .

$f (- 2) = - 20 + 20 - 15$

Simplify by adding and subtracting.

Add $- 20$ and $20$ .

$f (- 2) = 0 - 15$

Subtract $15$ from $0$ .

$f (- 2) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = - 2$ is $- 15$ .

$y = - 15$

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

$f (- 3) = - 5 {(- 3)}^{2} - 10 \cdot - 3 - 15$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 5 \cdot 9 - 10 \cdot - 3 - 15$

Multiply $- 5$ by $9$ .

$f (- 3) = - 45 - 10 \cdot - 3 - 15$

Multiply $- 10$ by $- 3$ .

$f (- 3) = - 45 + 30 - 15$

Simplify by adding and subtracting.

Add $- 45$ and $30$ .

$f (- 3) = - 15 - 15$

Subtract $15$ from $- 15$ .

$f (- 3) = - 30$

The final answer is $- 30$ .

$- 30$

The $y$ value at $x = - 3$ is $- 30$ .

$y = - 30$

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 5 {(0)}^{2} - 10 \cdot 0 - 15$

Simplify the result.

Simplify each term.

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

$f (0) = - 5 \cdot 0 - 10 \cdot 0 - 15$

Multiply $- 5$ by $0$ .

$f (0) = 0 - 10 \cdot 0 - 15$

Multiply $- 10$ by $0$ .

$f (0) = 0 + 0 - 15$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 15$

Subtract $15$ from $0$ .

$f (0) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = 0$ is $- 15$ .

$y = - 15$

Replace the variable $x$ with $1$ in the expression.

$f (1) = - 5 {(1)}^{2} - 10 \cdot 1 - 15$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = - 5 \cdot 1 - 10 \cdot 1 - 15$

Multiply $- 5$ by $1$ .

$f (1) = - 5 - 10 \cdot 1 - 15$

Multiply $- 10$ by $1$ .

$f (1) = - 5 - 10 - 15$

Simplify by subtracting numbers.

Subtract $10$ from $- 5$ .

$f (1) = - 15 - 15$

Subtract $15$ from $- 15$ .

$f (1) = - 30$

The final answer is $- 30$ .

$- 30$

The $y$ value at $x = 1$ is $- 30$ .

$y = - 30$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 30 \\ - 2 & - 15 \\ - 1 & - 10 \\ 0 & - 15 \\ 1 & - 30 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, - 10)$

Focus: $(- 1, - \frac{201}{20})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{199}{20}$

$\begin{array}{c} x & y \\ - 3 & - 30 \\ - 2 & - 15 \\ - 1 & - 10 \\ 0 & - 15 \\ 1 & - 30 \end{array}$

Step 4

Graph y=-5x^2-10x-15

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