1. The problem is to explain turning points in the context of IGCSE mathematics. 2. A turning point on a graph is where the curve changes direction from increasing to decreasing or from decreasing to increasing. 3. The formula used to find turning points involves calculus: turning points occur where the first derivative of the function equals zero, i.e., $$f'(x) = 0$$. 4. Important rules: - At a turning point, the slope of the tangent to the curve is zero. - Turning points can be maxima (peak), minima (valley), or points of inflection. 5. For example, consider the function $$y = x^2 - 4x + 3$$. 6. Find the first derivative: $$f'(x) = 2x - 4$$ 7. Set the derivative equal to zero to find critical points: $$2x - 4 = 0$$ $$\cancel{2}x - \cancel{4} = 0$$ $$x = 2$$ 8. To determine if this is a maximum or minimum, check the second derivative: $$f''(x) = 2$$ Since $$f''(2) = 2 > 0$$, the point at $$x=2$$ is a minimum. 9. Find the y-coordinate of the turning point by substituting $$x=2$$ into the original function: $$y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$$ 10. Therefore, the turning point is at $$(2, -1)$$, which is a minimum point. Turning points are important because they show where the graph changes direction, helping us understand the shape and behavior of functions.