Optimal Graphing Window For Domain And Range

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Optimal Graphing Window for Domain and Range

When you're trying to figure out the domain and range of a function using a graphing calculator, picking the right viewing window is super important. If your window is too small, you might miss key parts of the graph. Too big, and you might not see the details you need. Let's dive into how to choose the best window, especially when dealing with rational functions like the one you mentioned.

Understanding the Function

Before we even touch the graphing calculator, let's break down the function: n~(x)=x2−81x2−11x+18\tilde{n}(x)=\frac{x^2-81}{x^2-11 x+18}.

Factoring

First, we need to factor both the numerator and the denominator. Factoring helps us find important features like zeros and vertical asymptotes.

  • Numerator: x2−81=(x−9)(x+9)x^2 - 81 = (x - 9)(x + 9)
  • Denominator: x2−11x+18=(x−2)(x−9)x^2 - 11x + 18 = (x - 2)(x - 9)

So, our function becomes: n~(x)=(x−9)(x+9)(x−2)(x−9)\tilde{n}(x)=\frac{(x - 9)(x + 9)}{(x - 2)(x - 9)}

Simplifying

Notice that we have a common factor of (x−9)(x - 9) in both the numerator and the denominator. We can cancel this out, but it's super important to remember that this creates a hole in the graph at x=9x = 9. After canceling, our simplified function is:

n~(x)=x+9x−2\tilde{n}(x)=\frac{x + 9}{x - 2}, with a hole at x=9x = 9.

Key Features to Consider

To choose the best viewing window, we need to consider several key features of the function:

1. Zeros

Zeros are the x-values where the function equals zero. These are the points where the graph crosses the x-axis. From our simplified function, n~(x)=x+9x−2\tilde{n}(x)=\frac{x + 9}{x - 2}, the zero is at x=−9x = -9.

2. Vertical Asymptotes

Vertical asymptotes occur where the denominator of the simplified function equals zero. In our case, the denominator is x−2x - 2, so we have a vertical asymptote at x=2x = 2.

3. Holes

As we noted earlier, there's a hole at x=9x = 9. This means the function is undefined at this point, but the graph will look continuous except for a tiny gap.

4. Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as xx approaches positive or negative infinity. For rational functions, we look at the degrees of the numerator and denominator.

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0y = 0.
  • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. In our simplified function n~(x)=x+9x−2\tilde{n}(x)=\frac{x + 9}{x - 2}, the degrees are equal (both are 1), so the horizontal asymptote is y=11=1y = \frac{1}{1} = 1.
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there might be a slant asymptote, but we don't need to worry about that for this problem).

5. Y-intercept

The y-intercept is the value of the function when x=0x = 0. Plugging x=0x = 0 into our simplified function, we get:

n~(0)=0+90−2=9−2=−4.5\tilde{n}(0)=\frac{0 + 9}{0 - 2} = \frac{9}{-2} = -4.5

So, the y-intercept is at y=−4.5y = -4.5.

Choosing the Right Viewing Window

Okay, now that we know all the key features, let's figure out the best viewing window for our graphing calculator. We need to see the zeros, vertical asymptotes, horizontal asymptotes, the y-intercept, and the hole. A good viewing window should include values around these key points.

Initial Window

Let's start with a window that covers the important x-values:

  • Xmin: -15 (to see the zero at -9)
  • Xmax: 15 (to see the vertical asymptote at 2 and the hole at 9)

For the y-values, we need to see the horizontal asymptote at y = 1 and the y-intercept at -4.5. So, let's try:

  • Ymin: -10 (to see the y-intercept)
  • Ymax: 10 (to see the horizontal asymptote)

This gives us an initial viewing window of:

  • Xmin: -15, Xmax: 15
  • Ymin: -10, Ymax: 10

Adjusting the Window

After graphing the function with this initial window, we can make some adjustments if needed. Here’s what to look for:

  • Are all the key features visible? Can you see the zero at x=−9x = -9, the vertical asymptote at x=2x = 2, and the hole at x=9x = 9? Is the y-intercept at y=−4.5y = -4.5 visible?
  • Is the horizontal asymptote clear? Does the graph flatten out as xx approaches positive and negative infinity?
  • Is the graph too cramped or too wide? If the graph looks cramped, increase the Xmax and Xmin values. If it looks too wide, decrease them.
  • Are the y-values appropriately scaled? If the graph goes off the top or bottom of the screen, adjust the Ymax and Ymin values.

Fine-Tuning

Let's say, after graphing with our initial window, we want to get a better look at the hole at x=9x = 9. We might adjust the window to focus more on that region. For example:

  • Xmin: 5, Xmax: 13
  • Ymin: 0, Ymax: 2

This zoomed-in view will help us see the behavior of the function around the hole more clearly. Remember, graphing calculators don't actually show the hole; they just skip over the point. But by zooming in, we can confirm its existence and approximate its coordinates.

Why the Given Options Might Not Be Ideal

The option provided in the question is:

  • Xmin: -10, Xmax: 10
  • Ymin: -10, Ymax: 10

While this window is a good starting point, it might not be the most appropriate because:

  • It barely shows the vertical asymptote at x=2x = 2. You'd see it, but not with much detail.
  • It doesn't give much space to see the behavior of the graph as x approaches positive or negative infinity. Seeing the horizontal asymptote clearly is valuable.
  • It includes the zero at x = -9 and the y-intercept at y = -4.5, which is good.
  • The hole at x=9 is near the edge of the screen but still visible.

However, with our previous analysis, we determined that X should be viewed from at least -15 to 15. In this way, we can see all aspects of the function properly.

Conclusion

Choosing the right viewing window on your graphing calculator is a critical skill for understanding functions. By factoring the function, identifying key features like zeros, vertical asymptotes, holes, and horizontal asymptotes, and then iteratively adjusting your window, you can get a clear and accurate picture of the function's behavior. Start with a broad view and then zoom in on areas of interest to fully explore the graph. Don't be afraid to experiment with different window settings until you find one that works best for you!