Question Graph the quadratic inequality $y < -2x^2 - 8x - 12$ . Determine which ordered pairs $(x, y)$ are solutions.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic inequality in the form $y < -2x^2 - 8x - 12$ .
2. We need to graph the inequality to visualize the set of solutions.
3. We need to determine if the given ordered pairs are solutions to the inequality.

STEP 2

To graph the inequality, we first need to graph the corresponding quadratic equation $y = -2x^2 - 8x - 12$ as a parabola. This will serve as the boundary of the inequality.

STEP 3

Find the vertex of the parabola. The vertex form of a parabola is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex. We can find the vertex by completing the square or using the vertex formula $h = -\frac{b}{2a}$ .

STEP 4

Calculate the value of $h$ using the vertex formula.
$h = -\frac{b}{2a} = -\frac{-8}{2 \cdot (-2)} = -\frac{-8}{-4} = 2$

STEP 5

Substitute $x = h$ into the quadratic equation to find $k$ .
$k = -2(2)^2 - 8(2) - 12$

STEP 6

Calculate the value of $k$ .
$k = -2(4) - 16 - 12 = -8 - 16 - 12 = -36$

STEP 7

Now we have the vertex $(h, k) = (2, -36)$ .

STEP 8

Determine the axis of symmetry of the parabola, which is the vertical line $x = h$ , so in this case, $x = 2$ .

STEP 9

Find the y-intercept by setting $x = 0$ in the quadratic equation.
$y = -2(0)^2 - 8(0) - 12$

STEP 10

Calculate the y-intercept.
$y = -12$

STEP 11

Plot the vertex $(2, -36)$ and the y-intercept $(0, -12)$ on a coordinate plane.

STEP 12

Since $a < 0$ (where $a = -2$ ), the parabola opens downwards. Sketch the parabola with the vertex and y-intercept, making sure it opens downwards.

STEP 13

Shade the region below the parabola to represent the set of solutions for the inequality $y < -2x^2 - 8x - 12$ .

STEP 14

To check if an ordered pair $(x, y)$ is a solution to the inequality, substitute the values of $x$ and $y$ into the inequality and see if it holds true.

STEP 15

Check if the ordered pair $(-2, -4)$ is a solution.
$-4 < -2(-2)^2 - 8(-2) - 12$

STEP 16

Calculate the right side of the inequality.
$-4 < -2(4) + 16 - 12$

STEP 17

Simplify the right side of the inequality.
$-4 < -8 + 16 - 12$
$-4 < -4$

STEP 18

Since $-4$ is not less than $-4$ , the ordered pair $(-2, -4)$ is not a solution.

STEP 19

Check if the ordered pair $(-3, -10)$ is a solution.
$-10 < -2(-3)^2 - 8(-3) - 12$

STEP 20

Calculate the right side of the inequality.
$-10 < -2(9) + 24 - 12$

STEP 21

Simplify the right side of the inequality.
$-10 < -18 + 24 - 12$
$-10 < -6$

STEP 22

Since $-10$ is less than $-6$ , the ordered pair $(-3, -10)$ is a solution.

STEP 23

Check if the ordered pair $(0, -15)$ is a solution.
$-15 < -2(0)^2 - 8(0) - 12$

STEP 24

Calculate the right side of the inequality.
$-15 < -12$

STEP 25

Since $-15$ is less than $-12$ , the ordered pair $(0, -15)$ is a solution.

STEP 26

Check if the ordered pair $(-4, -12)$ is a solution.
$-12 < -2(-4)^2 - 8(-4) - 12$

STEP 27

Calculate the right side of the inequality.
$-12 < -2(16) + 32 - 12$

STEP 28

Simplify the right side of the inequality.
$-12 < -32 + 32 - 12$
$-12 < -12$

STEP 29

Since $-12$ is not less than $-12$ , the ordered pair $(-4, -12)$ is not a solution.
Solutions to the inequality are the ordered pairs that satisfy the inequality when substituted into it. Based on our checks:
- $(-2, -4)$ is not a solution. - $(-3, -10)$ is a solution. - $(0, -15)$ is a solution. - $(-4, -12)$ is not a solution.

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