Let's analyze the quadratic inequality [tex]\(y < x^2 + 2x - 6\)[/tex] to understand the properties of its graph.
First, identify the quadratic function associated with the inequality:
[tex]\[ y = x^2 + 2x - 6 \][/tex]
1. Shading inside the parabola:
The inequality [tex]\( y < x^2 + 2x - 6 \)[/tex] indicates that [tex]\( y \)[/tex] is less than the value of the quadratic function. In the context of graphing, the region that satisfies the inequality will be shaded below the parabola. Therefore, the shading is inside (or below) the parabola.
2. Direction in which the parabola opens:
The coefficient of [tex]\( x^2 \)[/tex] in the quadratic function [tex]\( y = x^2 + 2x - 6 \)[/tex] is positive (specifically, the coefficient is 1). When the coefficient of [tex]\( x^2 \)[/tex] is positive, the parabola opens upwards. Therefore, the statement that the parabola opens down is false.
3. Vertex location:
To find the vertex of the parabola given by the quadratic function [tex]\( y = x^2 + 2x - 6 \)[/tex], we can use the vertex formula. The x-coordinate of the vertex (h) is given by:
[tex]\[ h = -\frac{b}{2a} \][/tex]
where [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex] and [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex].
For the function [tex]\( y = x^2 + 2x - 6 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
Substituting these values into the vertex formula:
[tex]\[ h = -\frac{2}{2 \cdot 1} = -1 \][/tex]
To find the y-coordinate of the vertex (k), substitute [tex]\( x = -1 \)[/tex] back into the quadratic function:
[tex]\[ k = (-1)^2 + 2(-1) - 6 = 1 - 2 - 6 = -7 \][/tex]
Therefore, the vertex of the parabola is [tex]\((-1, -7)\)[/tex], so the statement that the vertex is located at [tex]\((-1, -7)\)[/tex] is true.
4. Type of line for the parabola:
Since the inequality is [tex]\( y < x^2 + 2x - 6 \)[/tex], which uses a strict inequality ([tex]\(<\)[/tex] rather than [tex]\(\leq\)[/tex]), the boundary line of the parabola on the graph will be dashed. This distinguishes the values that are strictly less. Hence, the parabolic curve is not a solid line. Therefore, the statement that the parabola is a solid line is false.
### Conclusion:
- The shading is inside the parabola: True
- The parabola opens down: False
- The vertex is located at [tex]\((-1, -7)\)[/tex]: True
- The parabola is a solid line: False
