I'm trying to graph <span title="k(x)=(x^2-9)/(x^3+4x^2-x-4),">k(x)=(x^2-9)/(x^3+4x^2-x-...</span> but i think there's remainder of -8, so how do i do it?

Question

if you can please explain the steps to how to graph it

Answer 1

I think if you have like a ti-83 or 84 graphing calculator you can just plug it in.

that&#039;s usually what I do. You can also go to trace and then it&#039;ll tell you the exact coordinates.

hah, i dunno if i helped much.

Answer 2

First examine the expression. the function will be undefined when the denominator is 0 so:

x^3 + 4x^2 - x - 4 = 0

By inspection you can see that 1 is a solution: 1 + 4 - 1 - 4 = 0

(x - 1)(x^2 + 5x + 4) .... pretty easy to factor since you know (x - 1) and the other term must be (x^2 .. some x term ... + 4)

(x - 1)(x + 4)(x + 1)

x = 1, x = -4 and x = -1 are zeros for the denominator and therefore places where the function k is undefined.

Also look at where k = 0 and that will be x^2 - 9 = 0

So k = 0 when x = 3 or x = -3

So the function factored is: (x - 3)(x + 3)/[(x - 1)(x + 4)(x + 1)]

You now have four regions to examine:

-Infinithy to -4

-4 to -1

-1 to 1

1 to +Infinity

And also look at what happens when x goes to + or - Infinity

As x gets very large we can ignore the other terms except the highest powers of x so k = x^2/x^3 = 1/x wich goes to 0 as x goes to either + or - Infinity.

As x goes to -Infinity k =1/x so k will be negative so 0 will be approached from the negative side.

As x goes to +Infinity k will be positive so 0 will be approached from the positive side.

Now let us examine each region in turn.

From -Infinity to -4.

As x -&gt; -Infinity k -&gt; 0 from the negative side

As x -&gt; -4 k -&gt; -Infinity (numerator is + and all three denominator terms are -)

So in this range the function will start at -Infinity and be asymptotic to y = 0, move to the right and curve downward and then go to -Infinity as x goes to -4.

From -4 to -1.

We know that the function crosses the x-axis at - 3.

As x -&gt; -4 from the right side, the numerator is + and the denominator has two - terms and one + so it will also be +. This means the function will approach +Infinity as x -&gt; -4 from the right hand side.

As x -&gt; -1 from the left side, the nummerator is - and the denominator has two - and one + terms so it is positive. This means the function will approach -Infinity as x -&gt; -1 from the left hand side.

So in this range the function will be asymptotic to x = -4 at +Infinity, move down to cross the x-axis at x = -3 and then move down to be asymptotic to x = -1. Above the x-axis it will be concave upward and below the x-axis it will be concave downward.

From -1 to 1.

This will be on either ve all positive or all negative since the y is 0 at only x = -3 and x = 3. Just look at some point in this range and x = 0 is an easy one. Both numerator and denominator are positive so the function will be greater than 0.

In this range it will be U shaped and asymptotic to x = -1 and x = 1.

From 1 to +Infinity.

We know that the function crosses the x-axis at x = 3 so we just need to find out what it does as x -&gt; 1 from the right hand side and as x -&gt; +Infinity. We know the latter from above and that is 0 and that it is approached from positive k. So as x -&gt; +Infinity then k -&gt; 0 from positive k. As x -&gt; 1 from the right side, the numerator will be - and the denominator will have two positive and one negative term so k will go to -Infinity.

So in this range the function will be asymtotic to x = 1 at -Infinity, curve up to cross the x-axis at x = 3 and then curve back down and become asymptotic to the x-axis with k &gt; 0 as it does so.

This will give you what you need to sketch the general form of the graph without doing calculations. If you need more detail then this would have to be filled in by doing a lot of work.

Make a list of x-values for each range. Plug these values into x and calculate the k values and the plot the (x,k) pairs. You will not have a remainer since the fraction will be one number divided by another which is just a number.

Examples:

use x = 0 as I did above. k = -9/(-4) = 9/4

use x = 4: k = (16 - 9)/(64 + 64 - 4 - 4) = 7/120 = 0.058333

use x = -2: k = (4 - 9)/(-8 + 16 + 2 - 4) = -5/6 = -0.83333

You will notice that the k values agree with the general form we determined above.