Question 8: a) Sketch the graph of the function y = x(x + 2)(2 - x). b) Calculate the point of inflexion for the function y = x³ - 2x² + x + 1.

Here are the solutions for the given questions: a) Sketch the graph of the function y = x(x+2)(2-x). As an AI, I cannot directly draw a graph. However, I can provide a detailed description of how to sketch it, including all the necessary points and the general shape. Step 1: Find the x-intercepts. Set y=0: x(x+2)(2-x) = 0 This gives x=0, x+2=0 x=-2, and 2-x=0 x=2. The x-intercepts are at (-2,0), (0,0), and (2,0). Step 2: Find the y-intercept. Set x=0: y = 0(0+2)(2-0) = 0 The y-intercept is at (0,0). Step 3: Determine the end behavior. Expand the function to find the leading term: y = x(x+2)(2-x) = x(2x+4-x^2-2x) = x(-x^2+4) = -x^3+4x The leading term is -x^3. Since the degree is odd (3) and the leading coefficient is negative (-1): • As x , y - (the graph goes down on the right side). • As x -, y (the graph goes up on the left side). Step 4: Sketch the graph. 1. Plot the x-intercepts at (-2,0), (0,0), and (2,0). 2. Starting from the top-left (as x -, y ), draw the curve downwards. 3. Pass through the x-intercept (-2,0). 4. The curve will then turn and rise, passing through the y-intercept and x-intercept (0,0). 5. It will continue to rise to a local maximum, then turn downwards. 6. Pass through the x-intercept (2,0). 7. Continue downwards to the bottom-right (as x , y -). The graph will be a cubic curve that starts high on the left, dips below the x-axis, crosses the origin, rises above the x-axis, and then dips below the x-axis again as it moves to the right. b) Calculate the point of inflexion for the function y = x^3 - 2x^2 + x + 1. Step 1: Find the first derivative of the function. (dy)/(dx) = (d)/(dx)(x^3 - 2x^2 + x + 1) (dy)/(dx) = 3x^2 - 4x + 1 Step 2: Find the second derivative of the function. (d^2y)/(dx^2) = (d)/(dx)(3x^2 - 4x + 1) (d^2y)/(dx^2) = 6x - 4 Step 3: Set the second derivative to zero to find the x-coordinate of the point of inflexion. 6x - 4 = 0 6x = 4 x = (4)/(6) x = (2)/(3) Step 4: Substitute the x-coordinate back into the original function to find the corresponding y-coordinate. y = ((2)/(3))^3 - 2((2)/(3))^2 + (2)/(3) + 1 y = (8)/(27) - 2((4)/(9)) + (2)/(3) + 1 y = (8)/(27) - (8)/(9) + (2)/(3) + 1 To combine these terms, find a common denominator, which is 27: y = (8)/(27) - (8 × 3)/(9 × 3) + (2 × 9)/(3 × 9) + (1 × 27)/(1 × 27) y = (8)/(27) - (24)/(27) + (18)/(27) + (27)/(27) y = (8 - 24 + 18 + 27)/(27) y = (29)/(27) The point of inflexion is ((2)/(3), (29)/(27)). Last free one today — make it count tomorrow, or type /upgrade for unlimited.