Key fetures (that I think are helpful) are finding where the vertex (if applicable), x and y intercepts, and the horizontal and vertical and slanted assymtotes and holes in the fn are. Assume you have a function in the form (P(x))/(Q(x)). To find the x intercepts, reduce/simpify the fraction and set the numerator to 0 and solve. To find the y intercepts, use the reducd fraction and find the result when you replace all x's with zero. To find the vertical asymtotes, you reduce/simplify the fraction and set the deomenator to zero, the result is the equaton of the vertical asemtote which the function never crosses. then, to find the horizontal assymtotes, you have 2 options: if the degree if P(x)<Q(x) then the horiziontal assymtote is y=0. If degree P(x)>Q(x), divide the leading coeficient of P(x) by the leading coeficient of Q(x). I don't rmember how to find slant assymtotest. To find whether the line crosses the horizontal assymtote, set the reduced function equal to the value of the horizontal assymtote, if you get a true statment, that is where it crosses, if false, it doesn't cross. when you eliminated something in reducing the fractio, that thing you elimated was a hole. Set that equal to zero and solve for x value, to get y value, input that x value into the reduced function and solve fo y value of hole.
to graph, plot x and y intercepts. remember that the lines get close to but never cross the vertical assymtotest, but they can cross the horizontal assymtotes sometimes. remember to plot the hole and draw the assymtotes.
The graph in the picture given is obviously a function of f(x) = x^3 which answers whether the graph is a function or not. In addition, the answer the part on why it's a graph, it's because it passes the vertical line test.
