Graph f(x) = x³ - 3x² - 4. Find and mark all x-intercepts and show the general shape of the cubic.

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Let's graph the cubic function f of x equals x cubed minus 3 x squared minus 4. First, we'll identify the key features. The function has one x-intercept at approximately 3.38, where the graph crosses the x-axis. The y-intercept is at the point (0, -4). We also have critical points at x equals 0 and x equals 2, which are the local maximum and minimum respectively. The general shape of this cubic function falls to the left and rises to the right, with the characteristic S-shape of a cubic. To find the x-intercept, we need to solve the equation x cubed minus 3 x squared minus 4 equals zero. This is not easily factorizable, so we'll use a numerical approach. Let's test some values. When x equals 3, we get f of 3 equals 27 minus 27 minus 4, which equals negative 4. When x equals 4, we get f of 4 equals 64 minus 48 minus 4, which equals positive 12. Since f of 3 is negative and f of 4 is positive, and the function is continuous, there must be a root between 3 and 4 where the graph crosses the x-axis. Using numerical methods, we can determine that this x-intercept is approximately 3.38. To find the critical points of our cubic function, we need to calculate the derivative and set it equal to zero. The derivative of f of x equals x cubed minus 3 x squared minus 4 is f prime of x equals 3 x squared minus 6 x, which can be factored as 3x times x minus 2. Setting this equal to zero, we get x equals 0 or x equals 2. These are our critical points. When we evaluate the function at these points, we get f of 0 equals negative 4, and f of 2 equals negative 8. Looking at the graph, we can see that x equals 0 is a local maximum with a horizontal tangent line, and x equals 2 is a local minimum with a horizontal tangent line. The derivative graph crosses the x-axis at exactly these two points, confirming our calculations. Let's analyze the general shape and end behavior of our cubic function. Since the leading coefficient is positive, as x approaches positive infinity, f of x approaches positive infinity. And as x approaches negative infinity, f of x approaches negative infinity. This gives our cubic function its characteristic shape. Starting from the left, the function falls from negative infinity, reaches a local maximum at the point (0, -4), then decreases to a local minimum at the point (2, -8). After that, it crosses the x-axis at approximately x equals 3.38, and continues rising toward positive infinity. This S-shaped curve is typical of cubic functions with a positive leading coefficient. The function has exactly one x-intercept, which we calculated earlier to be approximately 3.38. To summarize our analysis of the cubic function f of x equals x cubed minus 3 x squared minus 4: We found that this function has exactly one x-intercept at approximately 3.38. We identified critical points at x equals 0, which is a local maximum with a value of negative 4, and at x equals 2, which is a local minimum with a value of negative 8. The general shape of the function shows it falling from negative infinity on the left, reaching its maximum at the origin, decreasing to its minimum at x equals 2, crossing the x-axis at x equals 3.38, and then rising toward positive infinity. This S-shaped curve is characteristic of cubic functions with a positive leading coefficient. Understanding these key features gives us a complete picture of the function's behavior across its domain.