Curve Sketching - Graph the following equation with precision

Question

Graph the following equation with precision. Show all calculations. 

 

x⁴ - 2x³ - x + 2

--------------------------

     x³ - 4x

 

the dotted line is division. Just made it easier to read. (Hopefully)

 

Help is much appreciated with the whole question!

Answer 1

First, we need to factor this function.  We can try factoring by grouping.

 

x⁴ - x - 2x³ + 2

______________ =

    x³ - 4x

 

 

 

x(x³ - 1) - 2(x³ - 1)

_________________   =

    x(x² - 4)

 

 

(x - 2)(x³ - 1)

______________  =

  x(x - 2)(x + 2)

 

 

(x - 2)(x - 1)(x² + x + 1)

_____________________  =

   x(x - 2)(x + 2)

 

 

We can cancel out (x - 2).   This means that x=2 is a hole in the graph.

 

(x - 1)(x² + x + 1)

________________  =

     x(x + 2)

 

 

Now that we factored the function, we can find all the intercepts and asymptotes.

 

To find the x-intercepts, set the numerator equal to zero and solve for x.  Since the quadratic factor gives you complex roots, the only x-intercept is  x=1.

 

To find y-intercept, set x=0 and evaluate the function.

 

To find the vertical asymptote, set the denominator equal to zero and solve for x.  Your vertical asymptotes will be x=0 and x=0 and x=-2

 

If we expand the simplified expression, we can see that the degree of the numerator is greater than the degree of the denonomintor.  Therefore, you will have a slant asymptote.

 

To find the slant asymptote, expand the simplified expression.  Divide the expanded numerator by the expanded denominator using long division process.  You will have a quotient, and remainder.

 

The equation of the slant asymptote is the quotient part in the division.

 

So lets review.

 

x-intercept = (1, 0)

Hole at x=2

Y-intercept: none

vertical asymptote:  x = 0        and      x = -2

Slant asymptote: Quotient part using long division

 

You now have all the information.  Use them to sketch an accurate curve.

Answer 2

You have, actually, two main steps here. 1).You will want to factor the equations in the numerator and the denominator to find your vertical asymptotes and any point discontinuities (i.e.holes). Holes will occur where there is a common factor in both the numerator and the denominator. The vertical asymptote will be any remaining factor in the denominator that would make that equation zero. 2).You will also notice that the numerator is a higher degree polynomial, so you will have no horizontal asymptote, but you will need to divide the numerator by the denominator to discover your oblique asymptote. Once you sketch these out, you should be able to pick some points to plug in to help determine the actual curves. Hope that helps!