Explore a diverse range of topics and get expert answers on IDNLearn.com. Discover prompt and accurate answers from our community of experienced professionals.
Sagot :
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.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
