Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
