To solve the inequality [tex]\( y > 2x^2 + 6x - 3 \)[/tex], let's break it down into clear steps:
### Step 1: Identify the equation of the parabola
First, recognize that the boundary of the region is given by the equation:
[tex]\[ y = 2x^2 + 6x - 3 \][/tex]
This is the equation of a parabola.
### Step 2: Determine the shading region
Next, we need to analyze where the shading occurs. The inequality [tex]\( y > 2x^2 + 6x - 3 \)[/tex] tells us that we are looking for the values of [tex]\( y \)[/tex] that are greater than those on the parabola defined by [tex]\( y = 2x^2 + 6x - 3 \)[/tex].
For values of [tex]\( y \)[/tex] that are greater than the parabola, the graph is shaded above the curve.
### Step 3: Determine the type of curve
In this inequality, we have [tex]\( y > \)[/tex] rather than [tex]\( y \ge \)[/tex]. This distinction is important because:
- If the inequality were [tex]\( y \ge 2x^2 + 6x - 3 \)[/tex], then the points on the parabola would be included, and we would use a solid line to indicate this.
- However, since the inequality is strictly [tex]\( y > 2x^2 + 6x - 3 \)[/tex], the points on the parabola are not included in the solution, which means the boundary should be a dotted or dashed line.
### Summary
- Shading: The graph is shaded above the curve [tex]\( y = 2x^2 + 6x - 3 \)[/tex].
- Curve type: The curve itself is dotted.
Therefore, for the inequality [tex]\( y > 2x^2 + 6x - 3 \)[/tex], you shade the region above the parabola, and the boundary curve is dotted.
