Question

Select the correct answer. A diagonal curve declines through (negative 2, 5), (negative 1, 3), (0, 1), (1, negative 1), (2, negative 3), and (3, negative 5) on the x y coordinate plane. The area below the curve is shaded. Which of the following inequalities is graphed on the coordinate plane? A. y ≤ − 2 ⁢ x + 1 B. y ≥ − 2 ⁢ x + 1 C. y < − 2 ⁢ x + 1 D.

Answer 1

To determine the correct inequality that represents the shaded area below the diagonal curve, we need to identify the equation of the line that passes through the two endpoints of the curve, which are (-2, 5) and (3, -5).

First, we need to find the slope of the line:

slope = (change in y) / (change in x)

slope = (-5 - 5) / (3 - (-2))

slope = -10 / 5

slope = -2

Next, we can use the point-slope form of the equation of a line to find the equation of the line:

y - y1 = m(x - x1), where m is the slope and (x1, y1) is one of the points on the line.

Using the point (-2, 5), we have:

y - 5 = -2(x - (-2))

y - 5 = -2(x + 2)

y - 5 = -2x - 4

y = -2x + 1

So the equation of the line passing through the endpoints of the curve is y = -2x + 1.

To determine which inequality represents the shaded area below the curve, we can test a point that is not on the line, such as (0, 1).

Plugging in (0, 1) into the equation of the line, we get:

1 = -2(0) + 1

1 = 1

Since 1 is equal to 1, the point (0, 1) is on the line. Therefore, the inequality that represents the shaded area below the curve is y < -2x + 1, since the points below the line satisfy this inequality.

So the correct answer is C. y < -2x + 1.