Answer:
The correct options that describe the inequality (y + 1)² > 6·(x - 2)
1) Because only the 'y' quantity is squared the curve is a parabola
2) The parabola opens right because 6 is in front of the 'x' quantity
3) The shading area is outside the parabola because (0, 0) satisfies the inequality
4) The values subtracted from 'x', and 'y' in the equation are the coordinates of the vertex
Explanation:
The given inequality graphed by Miranda is presented as follows;
(y + 1)² > 6·(x - 2)
By simplification, we get;
Given that the inequality is of the form (y - k)² > 4·p·(x - h), we have;
The curve is a parabola as only the 'y' quantity is squared
-k = 1
∴ k = -1
4·p = 6
∴ p = 6/4 = 3/2 = 1.5
p = 1.5
-h = -2
∴ h = 2
The vertex of the parabola = (h, k) = (2, -1)
The axis of symmetry = The x-axis
p = 1.5 > 0, therefore, the parabola opens right
Therefore, the parabola opens right because 6 is in front of the 'x' quantity
There is a strict inequality therefore the boundary is not included in the solution set
When x = 0 and y = 0, we have;
(y + 1)² > 6·(x - 2)
(0 + 1)² > 6·(0 - 2)
1² > 6 × (-2)
1 > -12, which satisfies the inequality and (0, 0) is on the left and below the vertex, therefore, the shaded area is outside the parabola
The vertex of the parabola given in the form, (y - k)² > 4·p·(x - h), is (h, k), which are the values subtracted from 'x', and 'y' in the equation
Therefore, the vertex of the given parabola, (y + 1)² > 6·(x - 2) = (2, -1)