Question 390783
First we'll look at part b. The x-intercepts of a rational function will be where x makes the numerator zero. In order for an x-intercept to exist, (x-r) (where r is the intercept) must be a factor of the numerator. And for one x-intercept to be of multiplicity 2, (x-r) for that intercept must be a factor twice. So if {{{r[1]}}} is the x-intercept of multiplicity 1 and {{{r[2]}}} is the intercept of multiplicity 2, then the numerator will have the following factors:
{{{(x-r[1])(x-r[2])(x-r[2])}}} or {{{(x-r[1])(x-r[2])^2}}}<br>
Next we'll look at part c. For a rational function to have two vertical asymptotes, there must be two numbers that make the denominator zero. If we call these two numbers {{{r[3]}}} and {{{r[4]}}}, then the denominator must have the factors:
{{{(x-r[3])(x-r[4])}}}<br>
Last of all we will look at part a. So far our function looks like:
{{{f(x) = ((x-r[1])(x-r[2])^2)/((x-r[3])(x-r[4]))}}}
I hope it is easy to see that if we were to multiply out the numerator and denominator, the highest power term in the numerator would be {{{x^3}}} and the highest power term in the denominator would be {{{x^2}}}. And if we divide {{{x^3}}} by {{{x^2}}} we would get just "x". This would give us an oblique asymptote with a slope of 1 (since the coefficient of x is the slope). But we do not want a slope of 1. To get a different slope we need for the coefficients of the highest power terms to be different. One way to get different coefficients would be to add a constant factor to the numerator or denominator that is not a 1. If you add the factor to the numerator then that number will become the slope of the oblique asymptote.<br>
So our function now looks like:
{{{f(x) = (a(x-r[1])(x-r[2])^2)/((x-r[3])(x-r[4]))}}}
All we need to do now is pick numbers for "a", {{{r[1]}}}, {{{r[2]}}}, {{{r[3]}}} and {{{r[4]}}}. In order to get the function you want, pick any number but 1 for "a" and make sure that {{{r[1]}}}, {{{r[2]}}}, {{{r[3]}}} and {{{r[4]}}} are all different numbers. As long as you follow the above rules your function will meet all the requirements of the problem <i>no matter what numbers you pick!</i><br>
As for the graphing, here are some key ideas:<ul><li>The vertical asymptotes will be {{{x = r[3]}}} and {{{x = r[4]}}}.</li><li>The graph will cross the x-axis at {{{r[1]}}}</li><li>The graph will have a relative maximum or minimum at the point ({{{r[4]}}}, 0). (If you do not know what a "relative maximum or minimum" means, then the graph will "bounce off" the x-axis at ({{{r[4]}}}, 0)). (This happens because of the multiplicity of 2.)</li><li>If x=0 is not a vertical asymptote, then the graph will have a y-intercept. Put a zero in for x and figure out the value of the function when x is zero to determine the y-intercept.</li><li>To find the equation of the oblique asymptote:<ol><li>Multiply out both the numerator and the denominator.</li><li>Use long division to divide the numerator by the denominator.</li><li>You should get something in the form of:
ax + b + expression/denominator
For large x values the expression/denominator fraction will approach zero. So the oblique asymptote will be the line:
y = ax + b
where "a" is the number you picked earlier and "b" is some other number (maybe even the same as "a") depending on how the division works out.</li></ol></li><li>Two x-intercepts, an oblique asymptote and possibly a y-intercept are not enough to figure out the whole graph. More points are needed. Pick some additional values for x and find the y value for each one. This will give you additional points. Keep finding additional points and plot them until you have a sense as to how the graph goes. Among the x values you try, I would suggest picking x values a little to the left and a little to the right of each of the two vertical asymptotes.</li></ul>