What is/are the solution(s) to the function -3x = – 4 + x2 ? ANSWER x = -4 x = 1

A point on a graph moves so that the function y = x2 + 3x – 4 describes how the vertical position y is related to the horizontal position x. What is the vertical position when the horizontal position is –1? ANSWER -6

3. Graph the function y = x2 + 3x – 4 ANSWER (On next slide)

Graph a quadratic inequality EXAMPLE 1 Graph a quadratic inequality Graph y > x2 + 3x – 4. SOLUTION STEP 1 Graph y = x2 + 3x – 4. Because the inequality symbol is >, make the parabola dashed. STEP 2 Test a point inside the parabola, such as (0, 0). y > x2 + 3x – 4 0 > 02 + 3(0) – 4 ? 0 > – 4 ✓

EXAMPLE 1 Graph a quadratic inequality So, (0, 0) is a solution of the inequality. STEP 3 Shade the region inside the parabola.

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the inequality. y > x2 + 2x – 8 ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the inequality. y < 2x2 – 3x + 1 ANSWER

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the inequality. y < – x2 + 4x + 2 ANSWER

Graph a system of quadratic inequalities EXAMPLE 2 Graph a system of quadratic inequalities Graph the system of quadratic inequalities. y < – x2 + 4 Inequality 1 y > x2 – 2x – 3 Inequality 2 SOLUTION STEP 1 Graph y ≤ – x2 + 4. The graph is the red region inside and including the parabola y = – x2 + 4.

EXAMPLE 2 Graph a system of quadratic inequalities STEP 2 Graph y > x2– 2x – 3. The graph is the blue region inside (but not including) the parabola y = x2 –2x – 3. STEP 3 Identify the purple region where the two graphs overlap. This region is the graph of the system.

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the system of inequalities consisting of y ≥ x2 and y < −x2 + 5. ANSWER

EXAMPLE 2 Use a quadratic inequality in real life A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided Rappelling W ≤ 1480d2 where d is the rope’s diameter (in inches). Graph the inequality. SOLUTION Graph W = 1480d2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).

EXAMPLE 2 Use a quadratic inequality in real life W ≤ 1480d2 2000 ≤ 1480(1)2 ? 2000 ≤ 1480 Because (1, 2000) is not a solution, shade the region below the parabola.