Question 1: Solve the inequality x² - 4x - 5 > 0 by constructing a sign diagram.

Answer: x < -1 or x > 5

Question 2: Plot the sign diagram for the quadratic inequality y = x² - 3x + 2 < 0.

Answer: Between x=1 and x=2

Question 3: An incorrect sign diagram shows the parabola opening upwards with the positive region shaded between the roots. Identify the mistake and correct it.

Answer: Mistake: The shading was between the roots for y > 0; for y < 0, the shading should be outside the roots.

Question 4: Construct a sign diagram for the inequality 2x² + 3x - 2 ≤ 0 and state which x-values satisfy it.

Answer: x between -1 and 0.5 inclusive

Question 5: A student incorrectly marks the roots at x = -2 and x = 1, but forgets to consider the sign of the quadratic. Explain how to determine the correct signs in the sign diagram.

Answer: Use a test point in each interval to determine if the quadratic is positive or negative in that interval.

Question 6: Given the quadratic y = x² + 6x + 9, draw its sign diagram and identify the solution to y > 0.

Answer: Always positive except at x = -3 where it equals zero

Question 7: A common misconception is that the parabola opens downwards for y = -x² + 4x + 1. Is this correct? Explain your reasoning.

Answer: No, because the coefficient of x² is negative, so it opens downward.

Question 8: Construct and analyze the sign diagram for y = x² - 2x - 3 > 0. Which x-values satisfy this inequality?

Answer: x < -1 or x > 3

Question 9: Explain why the sign diagram for y = x² + 4x + 4 ≥ 0 shows a parabola touching the x-axis at x = -2.

Answer: Because the quadratic is a perfect square, (x + 2)², which touches the x-axis at x = -2.

Question 10: A student claims that the sign diagram for y = x² - 5x + 6 < 0 is incorrect because they marked the roots at x=2 and x=4, but the signs are reversed. What is the correct sign pattern?

Answer: The parabola opens upward with roots at 2 and 3; between roots y<0, so the sign should be negative between 2 and 3.

Question 11: Determine the error in this sign diagram for y = x² - 7x + 10 < 0 where the roots are marked at x=2 and x=5, but the shaded region is outside the roots.

Answer: The parabola opens upward, so y<0 between roots. The shaded outside is incorrect; it should be shaded between x=2 and x=5.