Answer :
To graph the system of inequalities:
[tex]\[ \left\{ \begin{array}{l} y \geq 2x^2 + 2 \\ y \leq -\frac{1}{4}x - 2 \end{array} \right. \][/tex]
we will follow these steps:
### Step 1: Graph the Boundary Lines
First, we'll graph the boundaries of the inequalities, which are the equations:
1. [tex]\( y = 2x^2 + 2 \)[/tex]
2. [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]
### Step 2: Determine the Shaded Regions
Next, we will determine which regions to shade for each inequality:
1. For [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- This inequality includes the area above (or on) the parabola [tex]\( y = 2x^2 + 2 \)[/tex].
2. For [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- This inequality includes the area below (or on) the line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
### Step 3: Identify the Intersection Area
The solution to the system of inequalities is the region where the shaded areas for both inequalities overlap.
### Step 4: Draw the Graph
1. Graphing the Parabola [tex]\( y = 2x^2 + 2 \)[/tex]:
- The parabola opens upwards.
- The vertex of the parabola is at [tex]\( (0, 2) \)[/tex].
- To plot additional points, choose [tex]\( x \)[/tex]-values and solve for [tex]\( y \)[/tex]:
- [tex]\( x = 1 \)[/tex], [tex]\( y = 2(1)^2 + 2 = 4 \)[/tex]
- [tex]\( x = -1 \)[/tex], [tex]\( y = 2(-1)^2 + 2 = 4 \)[/tex]
- [tex]\( x = 2 \)[/tex], [tex]\( y = 2(2)^2 + 2 = 10 \)[/tex]
- [tex]\( x = -2 \)[/tex], [tex]\( y = 2(-2)^2 + 2 = 10 \)[/tex]
2. Graphing the Line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]:
- The slope is [tex]\( -\frac{1}{4} \)[/tex] and the y-intercept is [tex]\( -2 \)[/tex].
- To plot, start at the y-intercept ([tex]\(0, -2\)[/tex]) and use the slope:
- From [tex]\( (0, -2) \)[/tex], move 1 unit right and [tex]\(\frac{1}{4}\)[/tex] unit down (which is [tex]\(-0.25\)[/tex]), giving the point (4, -3), and repeat as necessary.
- Alternatively, chose [tex]\( x \)[/tex]-values and solve for [tex]\( y \)[/tex]:
- [tex]\( x = 4 \)[/tex], [tex]\( y = -\frac{1}{4}(4) - 2 = -3 \)[/tex]
- [tex]\( x = -4 \)[/tex], [tex]\( y = -\frac{1}{4}(-4) - 2 = -1 \)[/tex]
### Step 5: Shading the Regions
1. Shade the area above the parabola [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- This includes and goes up from the parabola [tex]\( y = 2x^2 + 2 \)[/tex].
2. Shade the area below the line [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- This includes and goes down from the linear line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
### Final Step: Marking the Intersection
- The feasible region that satisfies both inequalities will be where the shaded regions overlap.
### Graph Sketch
To summarize visually the steps above:
1. Draw the parabola [tex]\( y = 2x^2 + 2 \)[/tex]:
- Vertex at [tex]\( (0, 2) \)[/tex].
- Goes through points [tex]\((1, 4)\)[/tex], [tex]\((-1, 4)\)[/tex], [tex]\((2, 10)\)[/tex], [tex]\((-2, 10)\)[/tex].
2. Draw the line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]:
- Line passing through [tex]\((0, -2)\)[/tex], [tex]\((4, -3)\)[/tex], [tex]\((-4, -1)\)[/tex].
3. Shade above the parabola [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- Vertically upwards from along the parabola curve.
4. Shade below the line [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- Vertically downwards from the linear line.
5. Identify the Overlap:
- The overlap area represents the solution to the system of inequalities.
### Conclusion
By following these steps and visually interpreting the graphs, you can determine the solution region for the given system of inequalities.
[tex]\[ \left\{ \begin{array}{l} y \geq 2x^2 + 2 \\ y \leq -\frac{1}{4}x - 2 \end{array} \right. \][/tex]
we will follow these steps:
### Step 1: Graph the Boundary Lines
First, we'll graph the boundaries of the inequalities, which are the equations:
1. [tex]\( y = 2x^2 + 2 \)[/tex]
2. [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]
### Step 2: Determine the Shaded Regions
Next, we will determine which regions to shade for each inequality:
1. For [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- This inequality includes the area above (or on) the parabola [tex]\( y = 2x^2 + 2 \)[/tex].
2. For [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- This inequality includes the area below (or on) the line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
### Step 3: Identify the Intersection Area
The solution to the system of inequalities is the region where the shaded areas for both inequalities overlap.
### Step 4: Draw the Graph
1. Graphing the Parabola [tex]\( y = 2x^2 + 2 \)[/tex]:
- The parabola opens upwards.
- The vertex of the parabola is at [tex]\( (0, 2) \)[/tex].
- To plot additional points, choose [tex]\( x \)[/tex]-values and solve for [tex]\( y \)[/tex]:
- [tex]\( x = 1 \)[/tex], [tex]\( y = 2(1)^2 + 2 = 4 \)[/tex]
- [tex]\( x = -1 \)[/tex], [tex]\( y = 2(-1)^2 + 2 = 4 \)[/tex]
- [tex]\( x = 2 \)[/tex], [tex]\( y = 2(2)^2 + 2 = 10 \)[/tex]
- [tex]\( x = -2 \)[/tex], [tex]\( y = 2(-2)^2 + 2 = 10 \)[/tex]
2. Graphing the Line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]:
- The slope is [tex]\( -\frac{1}{4} \)[/tex] and the y-intercept is [tex]\( -2 \)[/tex].
- To plot, start at the y-intercept ([tex]\(0, -2\)[/tex]) and use the slope:
- From [tex]\( (0, -2) \)[/tex], move 1 unit right and [tex]\(\frac{1}{4}\)[/tex] unit down (which is [tex]\(-0.25\)[/tex]), giving the point (4, -3), and repeat as necessary.
- Alternatively, chose [tex]\( x \)[/tex]-values and solve for [tex]\( y \)[/tex]:
- [tex]\( x = 4 \)[/tex], [tex]\( y = -\frac{1}{4}(4) - 2 = -3 \)[/tex]
- [tex]\( x = -4 \)[/tex], [tex]\( y = -\frac{1}{4}(-4) - 2 = -1 \)[/tex]
### Step 5: Shading the Regions
1. Shade the area above the parabola [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- This includes and goes up from the parabola [tex]\( y = 2x^2 + 2 \)[/tex].
2. Shade the area below the line [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- This includes and goes down from the linear line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex].
### Final Step: Marking the Intersection
- The feasible region that satisfies both inequalities will be where the shaded regions overlap.
### Graph Sketch
To summarize visually the steps above:
1. Draw the parabola [tex]\( y = 2x^2 + 2 \)[/tex]:
- Vertex at [tex]\( (0, 2) \)[/tex].
- Goes through points [tex]\((1, 4)\)[/tex], [tex]\((-1, 4)\)[/tex], [tex]\((2, 10)\)[/tex], [tex]\((-2, 10)\)[/tex].
2. Draw the line [tex]\( y = -\frac{1}{4}x - 2 \)[/tex]:
- Line passing through [tex]\((0, -2)\)[/tex], [tex]\((4, -3)\)[/tex], [tex]\((-4, -1)\)[/tex].
3. Shade above the parabola [tex]\( y \geq 2x^2 + 2 \)[/tex]:
- Vertically upwards from along the parabola curve.
4. Shade below the line [tex]\( y \leq -\frac{1}{4}x - 2 \)[/tex]:
- Vertically downwards from the linear line.
5. Identify the Overlap:
- The overlap area represents the solution to the system of inequalities.
### Conclusion
By following these steps and visually interpreting the graphs, you can determine the solution region for the given system of inequalities.
