To find the rational roots of the given functions, use the Rational Root Theorem and trial and error. Divide the function by (x-a) using long division or synthetic division to find the quotient. Factor the quotient completely. Find the x-intercepts and y-intercept of the function and determine its left-end and right-end behavior. Plot the points obtained and connect them to graph the function.
The possible rational roots of each function can be found using the Rational Root Theorem. For f(x)=2x^3+3x^2-8x+3, the possible rational roots are ±1, ±3, ±1/2, and ±3/2. For f(x)=x^3-3x+2, the possible rational roots are ±1 and ±2. For f(x)=x^4-x^3-2x-4, the possible rational roots are ±1, ±2, and ±4.
To find one rational root x=a, you can use trial and error or synthetic division. For example, for f(x)=2x^3+3x^2-8x+3, you can start with the possible rational roots and test them using synthetic division until you find a root that gives a remainder of zero. Let's say x=1 is a root.
Using long division or synthetic division, divide f(x) by (x-a). For example, for f(x)=2x^3+3x^2-8x+3 divided by (x-1), you will get a quotient of 2x^2+5x-3.
To factor completely the quotient q(x), you can try factoring by grouping or using the quadratic formula if the quadratic factors cannot be easily factored. For example, the quadratic factors of q(x)=2x^2+5x-3 are (2x-1)(x+3).
The x-intercepts of f(x) are the values of x for which f(x)=0. You can find them by factoring out the roots from the factored form of f(x). For example, for f(x)=2x^3+3x^2-8x+3, the x-intercepts are x=1 and x=-3/2. The y-intercept is the value of f(x) when x=0. For example, for f(x)=2x^3+3x^2-8x+3, the y-intercept is f(0)=3.
The left-end behavior of f(x) can be determined by looking at the leading term of the function. For example, for f(x)=2x^3+3x^2-8x+3, as x approaches negative infinity, the leading term 2x^3 dominates and the function goes to negative infinity. The right-end behavior of f(x) can be determined by looking at the degree of the function and the sign of the leading coefficient. For example, for f(x)=2x^3+3x^2-8x+3, as x approaches positive infinity, the leading term 2x^3 dominates and the function goes to positive infinity.
To graph y=f(x), you can plot the x-intercepts, the y-intercept, and additional points obtained from the factored form of f(x). For example, for f(x)=2x^3+3x^2-8x+3, you can plot the points (1, 0), (-3/2, 0), and (0, 3) and connect them to get the graph.