solve and graph the inequalities
y >=xsquared
y<=x squared =3
Well, well, well! Time to put on my math clown shoes and tackle these inequalities. Let's start with the first one:
y ≥ x²
To graph this inequality, we'll need to draw a parabola that opens upward. The shaded area above the parabola represents all the possible values for y that satisfy the inequality.
Now, let's move on to the second one:
y ≤ x² + 3
To graph this one, we'll draw another parabola that opens upward, just like before. But this time, we need to shift the entire graph 3 units up on the y-axis. So, the shaded area below the parabola represents all the possible values for y that satisfy the inequality.
Remember, when you have a ≤ or ≥ sign in an inequality, the line on your graph should be solid, not dashed! We only use dashed lines when we have < or > signs.
Now put on your clown glasses and start graphing!
To solve and graph the inequalities y >= x^2 and y <= x^2 + 3, we will first solve them separately and then graph the solutions on a coordinate plane.
1. Solving y >= x^2:
To solve this inequality, we need to determine the region on the graph where y is greater than or equal to the square of x.
Step 1: Identify the boundary by setting y = x^2:
y = x^2
Step 2: Identify the inequality sign. Since it is y >= x^2, the boundary line should be solid (indicating that y is equal to x^2) and the region above the line will be shaded.
Step 3: Graph the boundary line y = x^2 and shade the area above the line.
Here is the graph for y >= x^2:
2. Solving y <= x^2 + 3:
To solve this inequality, we need to determine the region on the graph where y is less than or equal to the square of x plus 3.
Step 1: Identify the boundary by setting y = x^2 + 3:
y = x^2 + 3
Step 2: Identify the inequality sign. Since it is y <= x^2 + 3, the boundary line should be solid (indicating that y is equal to x^2 + 3) and the region below the line will be shaded.
Step 3: Graph the boundary line y = x^2 + 3 and shade the area below the line.
Here is the graph for y <= x^2 + 3:
3. Graphing both inequalities together:
To graph both inequalities together, we need to find the overlapping region of the two graphs.
Since the shaded regions above and below the boundary lines will overlap between y >= x^2 and y <= x^2 + 3, the common shaded region will be the solution to both inequalities.
Here is the graph showing the overlapping solution for both inequalities:
The common shaded region represents the solution to the system of inequalities y >= x^2 and y <= x^2 + 3.
To solve and graph the inequalities, follow these steps:
1. Solve the first inequality, y >= x^2:
- Start by setting y equal to x^2.
- Since y is greater than or equal to x^2, we can draw a solid line (not dashed) on the graph to represent this.
- Next, shade the region above the parabola, as the inequality is y >= x^2.
- This includes all points above the parabola and on the parabola itself.
2. Solve the second inequality, y <= x^2 + 3:
- Start by setting y equal to x^2 + 3.
- Since y is less than or equal to x^2 + 3, we can draw a solid line on the graph.
- Shade the region below the parabola, as the inequality is y <= x^2 + 3.
- This includes all points below the parabola and on the parabola itself.
By graphing both inequalities, you can see the overlapping region that satisfies both conditions. This area represents the solution to the system of inequalities.
To graph the inequalities, you can use a graphing calculator, graphing software, or plot the points manually.