Question One Carefully sketch the graph of the cubic function \( f(x)=x^{3}+3 x^{2}-x-3 \) clearl showing the intercepts and vertices \( [6 \mathrm{mks}] \)
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To sketch the graph of the cubic function \( f(x) = x^3 + 3x^2 - x - 3 \), start by finding the intercepts. Set \( f(x) = 0 \) to find the x-intercepts. The polynomial can be factored or solved using the Rational Root Theorem, leading to roots at \( x = 1 \) and \( x = -3 \). The y-intercept is found by evaluating \( f(0) \), giving \( f(0) = -3 \). Next, identify the critical points by taking the derivative: \( f'(x) = 3x^2 + 6x - 1 \). Setting this equal to zero, you can find the x-values where the function changes direction, which leads to vertices. The nature of the graph can be inferred by evaluating the second derivative \( f''(x) = 6x + 6 \) to determine concavity. Plot these points and sketch smoothly connecting curves to capture the behavior of the function as \( x \) approaches the intercepts and as it moves towards positive and negative infinity. Remember to label your axes and indicate the calculated points for clarity! Analyzing the function reveals that cubic functions will exhibit a classic "S" shape, but the interplay between the roots and critical points can give each graph its unique flair!