Question 1: Using the graph, estimate the gradient of the curve at the point where x = 2 by drawing a tangent and calculating the rise over run.

Answer: Calculate the change in y over change in x from the tangent line.

Question 2: Plot the graph of y = x^2 for x between -3 and 3. Estimate the gradient at x = 1 by drawing a tangent line.

Answer: Estimate using the drawn tangent; approximate gradient ~2.

Question 3: On the provided grid, construct a tangent to y = √x at x = 4 and estimate its gradient.

Answer: Estimate the slope visually from the tangent line.

Question 4: A curve passes through the points (1, 2) and (3, 8). Estimate the gradient at x = 2 by drawing a suitable tangent.

Answer: Calculate the gradient of the secant line; estimate local gradient.

Question 5: Estimate the gradient of y = 3x + 1 at x = 0 by drawing a tangent on the graph.

Answer: Since y=3x+1 is linear, gradient = 3.

Question 6: Given the graph of a curve, estimate the gradient at a point where the curve is steep and at a point where it is flat. Describe your method.

Answer: Use tangent lines and compare slopes visually; steeper means larger gradient, flatter means smaller.

Question 7: Plot the graph of y=|x| and estimate the gradient at x=0 by drawing tangents on both sides.

Answer: Left side: slope ≈ -1; right side: slope ≈ 1.

Question 8: A quadratic curve passes through points (-1, 1) and (2, 9). Estimate the gradient at x=1 by drawing a tangent.

Answer: Estimate from the tangent line; approximate gradient ~4.

Question 9: Construct a tangent to y = x^3 at x = 2 and estimate the gradient.

Answer: Estimate the slope using the tangent; approximate gradient ~12.

Question 10: Identify and correct the mistake in estimating the gradient of a curve if a student drew a very flat tangent when the curve was steep.

Answer: Mistake: Incorrectly drawn tangent; should have aligned visually with the steepness of the curve at that point.