Question 1: Explain in your own words what a sign diagram shows for a quadratic inequality.

Answer: A sign diagram shows where the quadratic expression is positive or negative by dividing the number line at its roots.

Question 1: Find the roots of the quadratic equation x^2 - 5x + 6 = 0.

Answer: x=2, x=3

Question 2: Construct the sign diagram for the inequality x^2 - 5x + 6 > 0.

Answer: Positive outside the roots 2 and 3 (i.e., for x<2 and x>3).

Question 3: Determine where the quadratic y = x^2 - 4x + 3 is negative.

Answer: Between the roots 1 and 3.

Question 4: Plot the graph of y = -x^2 + 4x - 3 and identify the intervals where y > 0.

Answer: y>0 between the roots, which are at x=1 and x=3.

Question 5: Solve the inequality x^2 + 2x - 8 ≤ 0 using a sign diagram.

Answer: Between the roots x=-4 and x=2.

Question 6: A ball is thrown upwards, and its height h after t seconds is modeled by h= -5t^2 + 20t. Use a sign diagram to find when the ball is above 0 meters.

Answer: Ball is above 0 meters between t=0 and t=4 seconds.

Question 7: Construct and interpret the sign diagram for y = 3x^2 - 9x.

Answer: Roots at x=0 and x=3; positive outside the roots, negative between.

Question 8: A manufacturer finds that the cost C (in dollars) to produce x items is given by C= 2x^2 - 12x + 20. When is the cost less than $40? Use a sign diagram.

Answer: Cost less than $40 when x is between 1 and 5.

Question 9: Determine the intervals where the quadratic y = x^2 + 4x + 3 is negative, and draw its sign diagram.

Answer: Negative between the roots x=-3 and x=-1.

Question 10: Identify and correct the mistake in this student’s sign diagram: They marked the quadratic y = x^2 - 4x + 4 as positive everywhere.

Answer: The quadratic is a perfect square, y=(x-2)^2, so it is zero at x=2 and positive elsewhere, but not positive everywhere.