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Understanding critical points is crucial for graphing cubic functions and analyzing their behavior. These points on the graph are where the function switches direction, meaning it changes from increasing to decreasing, or vice versa. They're where the slope of the tangent (derivative) equals zero or where the derivative does not exist.

In the context of the function \(y=x^3-x\), we find the derivative of the function, \(y'(x)=3x^2-1\), to locate these points. We then set the derivative equal to zero to compute the critical points. For \(y=x^3-x\), the critical points are where \(x=\pm \sqrt{\frac{1}{3}}\). It's essential to not only calculate these values but also to determine whether these points are maxima or minima by using the first or second derivative test or by evaluating the function's value at these points.

Inflection points are points on the graph where the curvature changes sign. They provide insights into the concavity of the graph—whether it's curving upwards or downwards. To find these inflection points, the second derivative of the function is used.

Specifically, for the cubic function \(y=x^3-x\), the second derivative is \(y''(x)=6x\). Setting this equal to zero reveals that the only inflection point is at \(x=0\). At this point, the curvature of our graph switches from concave up (shaped like a cup) to concave down (shaped like a cap) or vice versa. In the exercise, the inflection point coincides with the origin, demonstrating both its importance and the simplicity often found in cubic function behavior.

X-intercepts are the points where the graph intersects the x-axis. These are the real roots of the function. In real-world scenarios, finding the x-intercepts can signify when a certain quantity reaches zero, which can have practical significance depending on the context of the problem.

For graphing \(y=x^3-x\), we set the output \(y\) to zero and solve the equation to find the x-intercepts. Here, factoring the equation gives us three solutions: \(x=-1, 0, 1\), indicating that the graph crosses the x-axis at three points: \((-1,0)\), \((0,0)\), and \((1,0)\). It is helpful to plot these points early in the graphing process as they provide a foundation around which the rest of the graph is constructed.

The y-intercept is a single point where the function crosses the y-axis. It's considered a function's baseline and is often easy to calculate by simply setting \(x=0\) in the function's equation. For many functions, this intercept serves as a starting or reference point for graphing the entire curve.

For the given cubic function \(y=x^3-x\), the y-intercept is found by computing \(y=0^3-0\), which simplifies to \(y=0\). Therefore, the graph intersects the y-axis at the origin \((0,0)\). Though it might seem trivial since both the x-intercept and y-intercept in this case are the same point, understanding that this isn't always the case is important for more complex functions.