Given the graph y = (x2 - 9)(x + 3).

Let's re-solve Question 17, Mazel. Question 17: Given the graph y = (x^2 - 9)(x + 3). a) The x-intercept of the curve Step 1: To find the x-intercept, set y=0. 0 = (x^2 - 9)(x + 3) Step 2: Factor the term (x^2 - 9) using the difference of squares formula, a^2 - b^2 = (a-b)(a+b). 0 = (x-3)(x+3)(x+3) 0 = (x-3)(x+3)^2 Step 3: Solve for x by setting each factor to zero. x-3 = 0 x = 3 x+3 = 0 x = -3 The x-intercepts are (3,0) and (-3,0). b) The y-intercept of the curve Step 1: To find the y-intercept, set x=0. y = (0^2 - 9)(0 + 3) Step 2: Calculate the value of y. y = (-9)(3) y = -27 The y-intercept is (0,-27). c) The stationary point of the curve Step 1: Expand the equation for y. y = (x^2 - 9)(x + 3) y = x^3 + 3x^2 - 9x - 27 Step 2: Differentiate y with respect to x to find the gradient function. (dy)/(dx) = (d)/(dx)(x^3 + 3x^2 - 9x - 27) (dy)/(dx) = 3x^2 + 6x - 9 Step 3: Set (dy)/(dx) = 0 to find the x-coordinates of the stationary points. 3x^2 + 6x - 9 = 0 Divide the entire equation by 3: x^2 + 2x - 3 = 0 Factor the quadratic equation: (x+3)(x-1) = 0 This gives two possible x-values: x+3 = 0 x = -3 x-1 = 0 x = 1 Step 4: Substitute these x-values back into the original curve equation y = x^3 + 3x^2 - 9x - 27 to find the corresponding y-coordinates. For x=-3: y = (-3)^3 + 3(-3)^2 - 9(-3) - 27 y = -27 + 3(9) + 27 - 27 y = -27 + 27 + 27 - 27 y = 0 So, one stationary point is (-3,0). For x=1: y = (1)^3 + 3(1)^2 - 9(1) - 27 y = 1 + 3 - 9 - 27 y = 4 - 36 y = -32 So, the other stationary point is (1,-32). The stationary points are (-3,0) and (1,-32). d) The nature of the stationary point in c above Step 1: Find the second derivative of the curve. (d^2y)/(dx^2) = (d)/(dx)(3x^2 + 6x - 9) (d^2y)/(dx^2) = 6x + 6 Step 2: Evaluate the second derivative at each stationary point's x-coordinate. For x=-3: (d^2y)/(dx^2)|_x=-3 = 6(-3) + 6 = -18 + 6 = -12 Since (d^2y)/(dx^2) < 0, the point (-3,0) is a maximum point. For x=1: (d^2y)/(dx^2)|_x=1 = 6(1) + 6 = 6 + 6 = 12 Since (d^2y)/(dx^2) > 0, the point (1,-32) is a minimum point. So, (-3,0) is a maximum point while (1,-32) is a minimum point. e) Sketch the curve. The sketch should show the following key features: • X-intercepts at (3,0) and (-3,0). • Y-intercept at (0,-27). • A local maximum point at (-3,0). • A local minimum point at (1,-32). The curve rises from the left, reaches a maximum at (-3,0), then decreases, passing through the y-intercept (0,-27), continues to decrease to the minimum point at (1,-32), and then increases to the right, passing through the x-intercept (3,0). The provided image shows a correct sketch of the curve. That's 2 down. 3 left today — send the next one.