Question 1: Plot the graph of y = 2x + 1 and y = -x² + 4 on the grid. Find their points of intersection.

Answer: Intersection points at approximately (1,3) and (-2, -3)

Question 2: Plot the quadratic y = x² - 4x + 3 and the linear y = 2x - 1. Determine whether the graphs intersect and if so, estimate the points.

Answer: Graphs intersect at approximately x = 1 and x = 3; points (1, 1) and (3, 5)

Question 3: On the grid, draw the linear equation y = -x + 2 and the quadratic y = x². Find their intersection points.

Answer: Intersections at approximately (0, 2) and (1.41, 1.99)

Question 4: Construct a quadratic graph that intersects the line y=3x+1 at exactly two points. Describe the shape and position of your parabola.

Answer: A parabola opening upwards, with vertex below the line, crossing y=3x+1 at two points

Question 5: Plot y = -x² + 4x and y = 2x - 3. Find their approximate intersection points and shade the region between the curves.

Answer: Intersections at approximately x=1 and x=3; region between x=1 and x=3 between the curves

Question 6: Given the graph of y = x² - 4x + 1 and y = -x + 2, determine the nature of their intersection.

Answer: Two points of intersection, one above and one below the x-axis

Question 7: Plot the equations y = x² - 2x and y = 4 - x. Find the intersection points and describe the solutions.

Answer: Intersections at x=1 and x=2; points (1, -1) and (2, 0)

Question 8: On the grid, draw the line y = 3x and the parabola y = x² + 2x. Find the approximate points where they intersect.

Answer: Intersections at approximately x=-1 and x=2; points (-1, -3) and (2, 6)

Question 9: A company models profit with y = -x² + 10x - 16 and costs with y = 4x + 2. Plot both and determine the maximum profit and the break-even points.

Answer: Maximum profit at x=5 with y=14; break-even points approximately at x=1 and x=4

Question 10: Identify and correct the mistake in the following graph: students plotted y = x² + 2x but marked it as a linear function.

Answer: The graph is quadratic, not linear; the curve opens upwards, not a straight line

Question 11: Challenge: Construct the graphs of y = x² - 6x + 9 and y = -2x + 4. Find their points of intersection and describe the geometric relationship.

Answer: Intersections at x=3; tangent point; parabola touches line at (3, -2)