Graphing The Function F(x) = (x^4 - 3x) / (x^2 - 9)

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Hey guys! Let's dive into graphing the function f(x) = (x^4 - 3x) / (x^2 - 9). This looks like a fun challenge, and we'll break it down step by step to make it super clear. We'll cover everything from identifying key features like asymptotes and intercepts to understanding the function's overall behavior. So, grab your pencils and let's get started!

Understanding the Function

Before we jump into graphing, let's really understand this function. Our function is a rational function, which means it's a fraction where both the numerator and denominator are polynomials. In this case, we have a polynomial of degree 4 in the numerator (x^4 - 3x) and a polynomial of degree 2 in the denominator (x^2 - 9). Rational functions can have some cool features, like vertical and horizontal asymptotes, which we'll need to identify.

First things first, let's look at the domain. The domain of a rational function is all real numbers except for the values that make the denominator zero. So, we need to find the values of x for which x^2 - 9 = 0. This factors to (x - 3)(x + 3) = 0, so the denominator is zero when x = 3 and x = -3. This means our function is undefined at these points, and we'll likely have vertical asymptotes there.

Next, let's think about intercepts. The y-intercept is the value of the function when x = 0. Plugging in x = 0 into our function, we get f(0) = (0^4 - 3(0)) / (0^2 - 9) = 0 / -9 = 0. So, the y-intercept is at the origin (0, 0). To find the x-intercepts, we need to find the values of x for which f(x) = 0. This happens when the numerator is zero, so we need to solve x^4 - 3x = 0. Factoring out an x, we get x(x^3 - 3) = 0. This gives us one x-intercept at x = 0 (which we already knew) and another where x^3 = 3, which means x = ∛3 (approximately 1.44). So, we have x-intercepts at (0, 0) and (∛3, 0).

Understanding these basics – the domain, the points where the function is undefined, and the intercepts – is crucial for sketching an accurate graph. We're building a solid foundation here, guys!

Identifying Asymptotes

Now, let's dig deeper into identifying asymptotes, which are like invisible guide rails that our function's graph will approach but never quite touch. Asymptotes are super important for understanding the overall behavior of the function, especially as x gets really large or really small.

We've already touched on vertical asymptotes. These occur where the denominator of the rational function is zero, and the numerator isn't zero at the same point. We found that our denominator, x^2 - 9, is zero at x = 3 and x = -3. Let's check the numerator at these points. When x = 3, the numerator is 3^4 - 3(3) = 81 - 9 = 72, which is not zero. When x = -3, the numerator is (-3)^4 - 3(-3) = 81 + 9 = 90, which is also not zero. So, we have vertical asymptotes at x = 3 and x = -3. These lines will be vertical on our graph, and the function will shoot off towards positive or negative infinity as x gets close to these values.

Next, let's consider horizontal or slant asymptotes. To find these, we need to compare the degrees of the polynomials in the numerator and denominator. Our numerator has degree 4 (x^4) and our denominator has degree 2 (x^2). When the degree of the numerator is greater than the degree of the denominator, we don't have a horizontal asymptote. Instead, we might have a slant asymptote (also called an oblique asymptote). Since the degree of the numerator is exactly two greater than the degree of the denominator, we know that the quotient will be a quadratic, so we can expect that the function doesn't have a linear asymptote.

To get a quadratic expression that approximates the function, we can perform polynomial long division. Dividing x^4 - 3x by x^2 - 9 gives us a quotient of x^2 + 9 with a remainder of 81 - 3x. We can write our function as:

f(x) = x^2 + 9 + (81 - 3x) / (x^2 - 9)

As x approaches infinity (positive or negative), the term (81 - 3x) / (x^2 - 9) approaches zero, because the denominator grows much faster than the numerator. This means that for large values of x, our function behaves like f(x) ≈ x^2 + 9. So, we have a curvilinear asymptote that follows the graph of the parabola y = x^2 + 9. This is a cool feature – our function will get closer and closer to this parabola as we move away from the origin.

Identifying these asymptotes gives us a skeleton for our graph. We know where the function will shoot off to infinity and what general shape it will take as x gets large. Awesome, right?

Analyzing the Behavior of the Function

Alright, guys, now let's get into analyzing the behavior of the function between and around those asymptotes and intercepts. This is where we really see the function's personality shine through! We'll use test points and some logical reasoning to figure out whether the graph is above or below the x-axis in different intervals.

First, let's break the number line into intervals based on our vertical asymptotes and x-intercepts. We have vertical asymptotes at x = -3 and x = 3, and x-intercepts at x = 0 and x = ∛3. This gives us the following intervals to consider:

  1. x < -3
  2. -3 < x < 0
  3. 0 < x < ∛3
  4. ∛3 < x < 3
  5. x > 3

Now, we'll pick a test point in each interval and evaluate the function. This will tell us whether the function is positive (above the x-axis) or negative (below the x-axis) in that interval.

  1. x < -3: Let's try x = -4. f(-4) = ((-4)^4 - 3(-4)) / ((-4)^2 - 9) = (256 + 12) / (16 - 9) = 268 / 7, which is positive. So, the graph is above the x-axis in this interval.
  2. -3 < x < 0: Let's try x = -1. f(-1) = ((-1)^4 - 3(-1)) / ((-1)^2 - 9) = (1 + 3) / (1 - 9) = 4 / -8 = -1/2, which is negative. So, the graph is below the x-axis in this interval.
  3. 0 < x < ∛3: Let's try x = 1. f(1) = (1^4 - 3(1)) / (1^2 - 9) = (1 - 3) / (1 - 9) = -2 / -8 = 1/4, which is positive. So, the graph is above the x-axis in this interval.
  4. ∛3 < x < 3: Let's try x = 2. f(2) = (2^4 - 3(2)) / (2^2 - 9) = (16 - 6) / (4 - 9) = 10 / -5 = -2, which is negative. So, the graph is below the x-axis in this interval.
  5. x > 3: Let's try x = 4. f(4) = (4^4 - 3(4)) / (4^2 - 9) = (256 - 12) / (16 - 9) = 244 / 7, which is positive. So, the graph is above the x-axis in this interval.

Putting this all together, we know how the function behaves in each interval. It shoots up or down towards the vertical asymptotes, crosses the x-axis at the intercepts, and follows the general trend of the y = x^2 + 9 parabola for large x values. We're really piecing the puzzle together now!

Sketching the Graph

Okay, team, it's sketching time! We've done the hard work of figuring out the function's key features, and now we get to put it all together visually. This is where everything clicks into place.

Let's start by drawing our asymptotes. We have vertical asymptotes at x = 3 and x = -3, so draw dashed vertical lines there. We also have the curvilinear asymptote y = x^2 + 9, which is a parabola opening upwards. It's helpful to sketch this lightly as a guide.

Next, plot the intercepts. We have intercepts at (0, 0) and (∛3, 0). Mark these points on your graph.

Now, use the information about the function's behavior in each interval to sketch the curves.

  1. For x < -3, the function is positive and approaches the vertical asymptote at x = -3 from above. It will curve upwards, getting closer to the y = x^2 + 9 parabola as x becomes more negative.
  2. For -3 < x < 0, the function is negative and approaches the vertical asymptote at x = -3 from below. It passes through the origin (0, 0).
  3. For 0 < x < ∛3, the function is positive and increases from the origin, crossing the x-axis at x = ∛3.
  4. For ∛3 < x < 3, the function is negative and approaches the vertical asymptote at x = 3 from below.
  5. For x > 3, the function is positive and approaches the vertical asymptote at x = 3 from above. It will curve upwards, getting closer to the y = x^2 + 9 parabola as x becomes more positive.

Connect the pieces, making sure your graph follows the asymptotes and passes through the intercepts. The graph will have separate branches, one in each interval defined by the vertical asymptotes. It's like a roller coaster ride with some exciting twists and turns!

Conclusion

And there you have it, guys! We've successfully graphed the function f(x) = (x^4 - 3x) / (x^2 - 9). We started by understanding the function's domain and intercepts, then we identified vertical and curvilinear asymptotes. Finally, we analyzed the function's behavior in different intervals and sketched the graph. This process might seem like a lot of steps, but each step gives us a crucial piece of the puzzle.

Graphing rational functions can be challenging, but with a systematic approach and a little practice, you'll become a pro in no time. Remember, the key is to break it down into manageable steps and understand the underlying concepts. Now go out there and graph some more functions! You got this! Remember to use these skills and knowledge to solve more problems and help others understand the beauty of mathematics. Happy graphing!