Graphing Quadratic Inequalities 9.6 Graphing Quadratic Inequalities

Sketch the graph of 6x + 5y ≥ 30 Remember how to graph a linear inequality? Sketch the graph of 6x + 5y ≥ 30 Write in slope- intercept form: y ≥ -6/5x + 6 This will be a solid line. Test a point. (0,0) 6(0) + 5(0) ≥ 30 0 ≥ 30 Not a solution. Shade the side that doesn’t include (0,0). 6 4 2 -6 -4 -2 2 4 6 8 -2 -4 -6

Rules for Graphing Quadratic Inequalities: 1. Sketch the graph of the parabola y<ax2+bx+c use ------ line for <, > use ______ line for 2. Test points inside the U shape and one outside the U shape. 3. Shade the region that contains the solution.

2 Check discriminant first for number of solutions b2-4ac (-2)2-4(3)(4) 4-48 -44 NO SOLUTION = Quadratic equation won’t work for x-axis crossings.

Next Step… y ≥ 3x2 – 2x + 4 Find a vertex, then make a table. -b 2a -(-2) 2 ∙ 3 2 6 The x-coordinate of the vertex is 1/3.

Plug in 1/3 to original equation to find y value: y ≥ 3x2 – 2x + 4 3(1/3)2 – 2(1/3) + 4 3(1/9) – 2/3 + 4 1/3 – 2/3 + 4

Make a table of values to find points left and right of the vertex. 3x2 – 2x + 4 y (x, y) -1 3(-1)2–2(-1) + 4 9 (-1, 9) Vertex (1/3, 11/3) 2 3(2)2 - 2(2) + 4 12 (2, 12) Now sketch the graph.

y ≥ 3x2 – 2x + 4 y 8 Solid line 6 4 2 x -6 -4 -2 -2 2 4 6 8 -4 -6 -8

Check Point Out (0,0) In (0,6) 2 2 NO Yes

Final Graph y 8 2 6 4 2 x -6 -4 -2 -2 2 4 6 8 -4 -6 -8

Pick a point to the right and one to the left Graph Try this one y > -x2 + 2x – 1 Find the vertex Pick a point to the right and one to the left Graph Shade (pick a point and shade the true side) (1, 0)

Dotted line Shade outside y x 8 6 4 (1,0) -6 -4 4 6 8 -4 -6 -8 2 -2 -2