Analyzing Rational Functions: Intercepts, Asymptotes, And Holes

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Hey there, math enthusiasts! Let's dive into the fascinating world of rational functions. Today, we're going to dissect the function q(x) = (x^6(x+1)^4(x+5)^4(x+2)^4) / (x^8(x+1)^4(x+2)^4(x-5)^2). Don't worry; it looks a bit intimidating at first, but we'll break it down step by step. We'll uncover its x-intercepts, vertical asymptotes, holes, and the y-intercept. Get ready to flex those math muscles! Let's get started!

Finding the X-Intercepts of the Function

So, first things first, let's tackle those x-intercepts. Remember, guys, x-intercepts are simply the points where the graph of the function crosses the x-axis. At these points, the value of y or q(x) is always zero. To find them, we need to set the function equal to zero and solve for x. However, for rational functions, this simplifies to just finding the zeros of the numerator (the top part of the fraction) and making sure those values don't make the denominator (the bottom part) also zero. If they do, then we don't have an x-intercept but a hole. Let's get to it.

Our function is q(x) = (x^6(x+1)^4(x+5)^4(x+2)^4) / (x^8(x+1)^4(x+2)^4(x-5)^2). First, let's focus on the numerator: x^6(x+1)^4(x+5)^4(x+2)^4. Setting this equal to zero gives us the potential x-intercepts. We can see that the numerator becomes zero when: x = 0, x = -1, x = -5, and x = -2. Now we need to check the denominator to see if any of these values cause it to be zero as well. The denominator is: x^8(x+1)^4(x+2)^4(x-5)^2. If any of the values that make the numerator zero also make the denominator zero, then we have a hole, not an x-intercept. So, what do you know? The values x = -1 and x = -2 make the denominator equal to zero, since they are factors of the denominator. That means x = -1 and x = -2 are actually locations of holes in the graph of the function. The values x = 0 and x = -5 do not make the denominator zero, therefore, these are the x-intercepts of the function. Thus, the x-intercepts are at x = 0 and x = -5. Pretty cool, right?

Now, let's put on our thinking caps and consider the multiplicity of each zero. Multiplicity refers to how many times a particular factor appears in the factored form of the function. The exponent on the factor tells us the multiplicity. For x = 0, the factor is x^6, meaning the zero has a multiplicity of 6 (even). For x = -5, the factor is (x+5)^4, meaning the zero has a multiplicity of 4 (even). Even multiplicities tell us that the graph touches the x-axis at that intercept but doesn't cross it. This means that the function will touch the x-axis at x=0 and x=-5 but will not cross it. Keep this in mind when visualizing the graph of the function.

Determining the Vertical Asymptotes of the Function

Alright, time to switch gears and talk about vertical asymptotes. These are the invisible lines that the graph of our function approaches but never actually touches. They occur at the x-values where the denominator of the function is equal to zero, but the numerator is not. These are values where the function is undefined, and the graph shoots off to positive or negative infinity.

Recall the original function: q(x) = (x^6(x+1)^4(x+5)^4(x+2)^4) / (x^8(x+1)^4(x+2)^4(x-5)^2). The denominator is: x^8(x+1)^4(x+2)^4(x-5)^2. Let's figure out what makes this bad boy equal to zero. We can see that the denominator becomes zero when x = 0, x = -1, x = -2, and x = 5. However, we have to remember that x = 0, x = -1, and x = -2 all cause the numerator to equal zero as well. This means these are not vertical asymptotes, but the location of holes (as we figured out earlier). The only value that makes the denominator zero but not the numerator is x = 5. Therefore, the vertical asymptote is at x = 5.

So, we have one vertical asymptote at x = 5. It's super important to remember that the vertical asymptotes represent the values of x where the function is undefined and the graph shoots off to infinity (or negative infinity). When drawing the graph, you'll draw a dashed vertical line at x = 5 to represent this asymptote. It acts as a guide for the curve of the graph, telling you where it will go up or down, but never touch. This happens because the denominator approaches zero, while the numerator approaches a non-zero value.

Identifying Holes in the Function

Let's talk about holes. Holes in a rational function are like tiny gaps in the graph. They occur when a factor in the numerator and denominator cancels out. This means that a value of x makes both the numerator and denominator of the function equal to zero, but the function is still defined everywhere else.

Looking back at our function: q(x) = (x^6(x+1)^4(x+5)^4(x+2)^4) / (x^8(x+1)^4(x+2)^4(x-5)^2). We've already identified that x = -1 and x = -2 cause both the numerator and denominator to be zero. This is a clue that we have holes at these points. To find the exact coordinates of the holes, we need to simplify the function by canceling out the common factors in the numerator and denominator and plugging the x-values that create holes into the simplified function to find the corresponding y-value. First, let's simplify the function by cancelling out the common factors:

q(x) = (x^6(x+1)^4(x+5)^4(x+2)^4) / (x^8(x+1)^4(x+2)^4(x-5)^2)
q(x) = (x^6(x+5)^4) / (x^8(x-5)^2)
q(x) = ((x+5)^4) / (x^2(x-5)^2)

Now that the function has been simplified, we can find the coordinates of the holes. First, consider the x-value x = -1. Substitute -1 into the simplified function for x: q(-1) = ((-1)+5)^4 / ((-1)^2((-1)-5)^2). This simplifies to q(-1) = 4^4 / (1*(-6)^2) = 256/36 = 64/9. So, one hole is at the point (-1, 64/9). Now, consider the x-value x = -2. Substitute -2 into the simplified function for x: q(-2) = ((-2)+5)^4 / ((-2)^2((-2)-5)^2). This simplifies to q(-2) = 3^4 / (4*(-7)^2) = 81/196. Thus, the second hole is at the point (-2, 81/196). Therefore, the holes are at the points (-1, 64/9) and (-2, 81/196). So, we now know where our graph has these small gaps. These are crucial points to mark when you're sketching the graph of this function.

Finding the Y-Intercept of the Function

Finally, let's find that y-intercept! The y-intercept is the point where the graph crosses the y-axis. This happens when x = 0. To find it, we substitute x = 0 into our simplified function (to avoid any headaches with the undefined form). Remember the simplified form of our function is: q(x) = ((x+5)^4) / (x^2(x-5)^2). Now we plug in x = 0: q(0) = ((0+5)^4) / ((0)^2((0)-5)^2). However, this leads to division by zero, which tells us that there is no y-intercept. This is another feature that can influence how we understand the graph of this function. When you look at the plot, it will never intersect the y-axis!

Summary

Let's recap what we've discovered:

  • X-intercepts: The function has x-intercepts at x = 0 and x = -5.
  • Vertical Asymptotes: There is a vertical asymptote at x = 5.
  • Holes: The function has holes at the points (-1, 64/9) and (-2, 81/196).
  • Y-intercept: There is no y-intercept.

And there you have it! You've successfully dissected this rational function and uncovered its key characteristics. Congrats, guys! Keep practicing and exploring; the world of math is full of amazing things to discover. Remember that understanding intercepts, asymptotes, and holes provides a solid foundation for grasping the behavior and overall shape of any rational function. Happy calculating!