Graph the Inequality. y≤1/4 x^2 -1/2x+5/4. plot the vertex and four additional points, two on each side of the vertex. Shade in the region where it... |

Graph the inequality `y<= 1/4 x^2-1/2x+5/4` :

(1) The inequality is inclusive (less than or equal to) so the graph is a solid line. (If the inequality was strictly less than then you would use a dashed line)

(2) We graph the underlying equality `y=1/4 x^2 -1/2x+5/4` :

The axis of symmetry is the line `x=(-b)/(2a)` , so:

`x=(1/2)/(2(1/4))=(1/2)/(1/2)=1`

Evaluating the function at 1 yields:

`y=1/4(1)^2-1/2(1)+5/4=1/4-2/4+5/4=4/4=1`

Since the vertex lies on the axis of symmetry, the vertex is at (1,1)

(2) We evaluate the function at some points near 1:

If x=2 then `y=1/4(2)^2-1/2(2)+5/4=1-1+5/4=5/4` . Thus the point `(2,5/4)` lies on the graph. By symmetry, `(0,5/4)` also lies on the graph. (Note that evaluating the function at 0 yields `5/4` , which helps confirm the symmetry.)

If x=3 then `y=1/4(3)^2-1/2(3)+5/4=9/4-6/4+5/4=8/4=2` so the points (3,2) and (-1,2) lie on the graph.

(3) The graph:

(4) In order to shade the correct 1/2-plane, we pick a point not on the curve; say (0,0). We then see if the inequality holds for that point. Here `0<=1/4(0)^2-1/2(0)+5/4=5/4` is true, so every point on the same side of the curve will also be a solution.

Thus you should plot the graph as shown with a solid line, and shade "outside" or "beneath" the curve; everything on the same side of the curve as the origin.