Question 1: Determine the intervals where the quadratic expression x² - 4x - 5 is positive. Draw the sign diagram.

Answer: x < -1 or x > 5

Question 2: Plot the graph of y = 2x² - 3x - 2 on the grid. Use it to identify where y is less than zero.

Answer: x between -0.33 and 3/2

Question 3: Construct a sign diagram for the inequality x² + 6x + 9 < 0.

Answer: No solutions, since x² + 6x + 9 is always ≥ 0

Question 4: Explain why the quadratic y = x² - 2x - 8 is negative between its roots. Use a sign diagram in your explanation.

Answer: Because the parabola opens upward, y is below zero between the roots -2 and 4, where the sign diagram shows negative values.

Question 5: Given the quadratic inequality x² - 3x + 2 ≤ 0, identify all solutions and draw the sign diagram.

Answer: x in [1, 2], with sign diagram showing negative or zero between roots.

Question 6: A ball is thrown such that its height h(t) = -2t² + 4t + 1. When is the ball above 0? Use a sign diagram.

Answer: When t is between 0.19 and 2 seconds, as indicated by the sign diagram showing positive y-values.

Question 7: Construct a sign diagram for the inequality 3x² - 12x > 0.

Answer: x < 0 or x > 4

Question 8: Identify the mistake in this sign diagram: the graph shows y < 0 for x > 1, but the quadratic is x² - 4x + 3. Correct the diagram and explain.

Answer: The quadratic factors as (x-1)(x-3), negative between roots 1 and 3, so the diagram should show negative in that interval, not for x > 1.

Question 9: Solve the inequality (x - 2)(x + 3) > 0 by constructing a sign diagram. What are the solution intervals?

Answer: x < -3 or x > 2

Question 10: A quadratic function is positive outside its roots. Given roots at -1 and 4, sketch the sign diagram and state the inequality it satisfies.

Answer: x < -1 or x > 4; the quadratic is > 0 outside the roots.