Place the steps to sketch the graph of a rational function in the appropriate
order.
(to make it easier to answer you can abbreviate the first letter since they're all different [f, u, s, c])

· find the function's zeros and vertical asymptotes, and plot them on a number line.

· use test numbers to find where the function is a positive and where it is negative.

· sketch the function's graph, plotting additional points as guides as negative.

· choose test numbers to t the left and right of each of these places, and find the value of the function at each test number.

  f, c, u, s

Step-by-step explanation:

The points of interest in graphing are the asymptotes and intercepts. (Maxima and minima are also useful, but may not be as easy to find.) So, the first step is ...

  1. Find the function's zeros and vertical asymptotes.

You need to know the sign of the function near each of the above points, so you can choose test numbers either side of those points. Using the test numbers, the function can be evaluated, and/or the sign determined. So, the next steps are ...

  2. Choose test numbers to the left and right of ...

  3. Use test numbers to find [the sign of the function]

Finally, armed with this information, you can ...

  4. Sketch the function's graph ...

In summary, the steps are ...

  F, C, U, S

Additional comment

Whether the sign changes at an asymptote or root is determined by its multiplicity. An asymptote or root of odd degree will be a place where the sign changes. If the degree is even, the sign will be the same on either side of the asymptote or root.

This means that if you determine multiplicity in the first step (F), you can essentially skip the 2nd and 3rd steps by looking at the sign of the leading term of the function. That sign will determine the function value for large positive values of the variable, and you can track sign changes as you go left across each of the points you found in the first step.

- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form

18

step-by-step explanation:

draw the shape and you will see that f and k are actually at adjacent vertices (points of the hexagon that are neighboring).   so find the distance between (1,2) and (4,2).   that is easy since (1,2) and (4,2) is just a vertical distance to count, which is 3.   it is a regular polygon which means all sides are congruent.   you have 6 sides (hexa) so multiply 6 by 3 to receive 18.