Question 1: Plot the graph of y = 2x + 1 on the grid. Identify two points for this line.

Answer: e.g., (0,1) and (2,5)

Question 2: Plot the quadratic equation y = x^2 - 4 on the same grid. Mark its vertex.

Answer: Vertex at (0,-4)

Question 3: Find the coordinates where the line y = 2x + 1 intersects the parabola y = x^2 - 4. Show your working.

Answer: x = -3 and x = 1; points (-3,-5) and (1,-3)

Question 4: Describe the graphical solution to the system: y = 2x + 1 and y = x^2 - 4. What do the intersection points represent in real-world terms?

Answer: The points where the line and parabola intersect represent solutions where both conditions are true, such as break-even points or optimal solutions in real-world contexts.

Question 5: Plot the line y = -x + 3 and the parabola y = (x - 1)^2 on the same graph. Find their intersection point.

Answer: Intersection at (1,2)

Question 6: A car rental company charges a fixed fee plus per mile driven. If the total cost y is linear with respect to miles x, and the quadratic models fuel consumption y = 0.05x^2 - x + 20, plot both on the grid. Find where the costs are equal.

Answer: Solve 0.05x^2 - x + 20 = fixed fee; for example, if fixed fee is 25, then 0.05x^2 - x + 20 = 25; solution x ≈ 1.4 miles.

Question 7: Construct a graph to show y = 3x + 2 and y = -x^2 + 4. Clearly mark the intersection points.

Answer: Intersections at approximately (-1, -1) and (2, 8)

Question 8: Explain why the solution points on the graph are important in a real-world scenario, such as planning or budgeting.

Answer: They show where two conditions are met simultaneously, helping in decision-making, like finding cost points where expenses and savings balance.

Question 9: Identify and correct the common mistake: A student plots y = x^2 + 1 and y = 2x + 3 but claims they don't intersect. Check the calculations and explain.

Answer: They should solve x^2 + 1 = 2x + 3, leading to x^2 - 2x - 2 = 0; the solutions are x = 1 ± √3, so intersection points exist.

Question 10: Challenge: On the grid, draw the line y = -2x + 4 and the parabola y = x^2 - 4x + 3. Find all intersection points.

Answer: Intersections at (1, 2) and (3, -2)