which of the following is the graph of this inequality? \n$y \\geq -x^2 + 5x - 6$
which of the following is the graph of this inequality? \n$y \\geq -x^2 + 5x - 6$
Answer
Explanation:
Step1: Analyze the parabola equation
First, let's rewrite the quadratic function ( y = -x^2 + 5x - 6 ). We can factor it: ( y=-(x^2 - 5x + 6)=-(x - 2)(x - 3) ). So the roots of the parabola (where ( y = 0 )) are ( x = 2 ) and ( x = 3 ). The coefficient of ( x^2 ) is -1, which is negative, so the parabola opens downward.
Step2: Analyze the inequality ( y\geq -x^2 + 5x - 6 )
The inequality ( y\geq ) (the quadratic function) means we shade the region above or on the parabola. Since the parabola opens downward, the region above the parabola (including the parabola itself) is the solution set. Now let's check the graphs:
- The first graph: The parabola is solid (since the inequality is ( \geq ), the boundary is included), and the shading is outside the parabola? Wait, no. Wait, the parabola opens downward, so the "inside" (the region above the parabola) should be shaded. Wait, let's check the vertex. The vertex of ( y=-x^2 + 5x - 6 ) has x-coordinate ( x = -\frac{b}{2a}=-\frac{5}{2(-1)}=\frac{5}{2}=2.5 ). The y-coordinate at ( x = 2.5 ) is ( y=-(2.5)^2+5(2.5)-6=-6.25 + 12.5 - 6=0.25 ). So the vertex is at (2.5, 0.25).
Now, the inequality ( y\geq -x^2 + 5x - 6 ) means all points (x,y) where y is greater than or equal to the parabola. Since the parabola opens downward, the region above the parabola (including the parabola) is the solution. So the boundary (the parabola) should be solid (because of ( \geq )), and the shading should be above the parabola.
Looking at the three graphs:
- First graph: The parabola is solid, and the shading is outside the parabola? Wait, no. Wait, the first graph's shading is the pink area. Wait, the parabola opens downward, so the area above the parabola (the region where y is larger) would be the area that is above the curve. Wait, maybe I got it reversed. Let's take a test point. Let's take x = 0. Then the quadratic function at x=0 is ( y=-0 + 0 - 6=-6 ). The inequality is ( y\geq -6 ). So at x=0, y should be greater than or equal to -6. Let's check the graphs.
First graph: At x=0, the y-axis. The shading in the first graph: is (0,y) in the shaded area? The first graph's shading is the pink area. At x=0, the pink area is above y=-6? Wait, maybe the first graph is correct? Wait, no. Wait, the second graph has a dashed parabola, but our inequality is ( \geq ), so the boundary should be solid. The third graph has shading below the parabola. Wait, no. Wait, let's re-express:
The inequality ( y\geq -x^2 + 5x - 6 ) is equivalent to ( y + x^2 - 5x + 6\geq 0 ). The parabola is the boundary. Since the parabola opens downward, the solution set is the region that is on or above the parabola (because for a downward opening parabola, the inequality ( y\geq ) the parabola) will be the region that is above the parabola (since the parabola is the "ceiling" here). Wait, maybe the first graph is correct. Wait, the first graph has a solid parabola (correct, because of ( \geq )) and the shading is the area that includes the region above the parabola. Wait, let's check the vertex. The vertex is at (2.5, 0.25), so the parabola is above the x-axis at the vertex. So the region ( y\geq ) the parabola would include all points above the parabola, including the parabola. So the first graph: the parabola is solid, and the shading is the area that is above the parabola (since the parabola opens downward, the area outside the "cup" is the area above). Wait, maybe I was confused earlier. Let's check the three graphs:
- First graph: Solid parabola, shading is the area that is outside the parabola (the area that is above the parabola, since the parabola opens downward). The second graph has a dashed parabola (incorrect, since the inequality is ( \geq ), so boundary is solid). The third graph has shading below the parabola (incorrect, because ( y\geq ) the parabola, so should be above). So the first graph is correct.
Wait, maybe I made a mistake. Let's re-express the inequality. Let's solve ( y\geq -x^2 + 5x - 6 ). Let's rewrite it as ( x^2 - 5x + 6 + y\geq 0 ). The parabola is ( y = -x^2 + 5x - 6 ), so the inequality is ( y\geq ) that, so the region where y is greater than or equal to the parabola. Since the parabola opens downward, the set of points above the parabola (including the parabola) is the solution. So the boundary (the parabola) is solid, and the shading is above the parabola. The first graph has a solid parabola and shading above (the pink area is the shading, which is the area that is above the parabola, since the parabola is the curve opening downward, so the pink area is outside the "U" shape, which is the area above the parabola). The second graph has a dashed parabola (wrong, because ( \geq ) requires solid line), and the third graph has shading below (wrong, because ( y\geq ) should be above). So the first graph is the correct one.