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Understanding the end behavior of polynomial functions can be simplified with the Leading Coefficient Test, which relies on the function's highest degree term. This term, composed of the leading coefficient and the highest exponent, determines how the graph behaves as the input values become very large in the positive or negative direction.

For a polynomial with an even leading power like in the function f(x)=x^{4}-2x^{3}+x^{2}, the end behavior will mimic the sign of the leading coefficient: if it's positive, both ends of the graph will point upwards, and if it's negative, both ends will point downwards. In our case, the leading coefficient is positive, so the graph rises to infinity as x moves towards both positive and negative infinity.

The x-intercepts of a polynomial function, also known as zeros or roots, are the points where the graph intersects the x-axis. To find these intercepts, we set the function equal to zero and solve for x.

For f(x)=x^{4}-2x^{3}+x^{2}, this process revealed intercepts at x=0 and x=1. However, there's a twist: both these intercepts are repeated roots, indicating that the graph only touches and 'bounces off' the x-axis at these points, rather than crossing it.

The y-intercept is found easily by evaluating the function when x is zero. In our function, setting x to zero gives f(0)=0, which means the graph crosses the y-axis at the origin. This single point can be a useful reference when sketching the graph of the polynomial.

Symmetry in graphs can greatly aid in understanding the shape of a function. A graph with y-axis symmetry means that for every point (x, y) on the graph, the point (-x, y) will also be on the graph. Origin symmetry means that for every point (x, y), the point (-x, -y) will be mirrored across the origin.

Our function f(x) lacks both y-axis and origin symmetry since neither f(-x)=f(x) nor f(-x)=-f(x) hold true. Recognizing symmetry can simplify graphing tasks considerably but requires a keen understanding of these principles.

Turning points of a polynomial graph are points where the graph changes direction from increasing to decreasing or vice versa. The number of possible turning points for a polynomial function is always less than its degree. Hence, for a fourth-degree polynomial like the one we have in f(x), there can be up to three turning points.

By plotting the function, we observed exactly three turning points, aligning with the mathematical theory. Turning points are critical to sketching the overarching shape of the graph and determining its concavity at various intervals.

Polynomial functions are algebraic expressions that consist of terms in the form a_nx^n where n is a non-negative integer and a_n is a coefficient. They can have various degrees, which is reflected in the function's highest power, and each degree brings distinct characteristics and graph shapes.

The degree heavily influences the graph's end behavior, number of roots, turns, and symmetry. Higher-degree polynomials, such as our fourth-degree function f(x), tend to have more complexity in their graphs, requiring a deeper analysis of points such as intercepts and turning behavior to accurately plot them.