Graphing F(x) = (5x)/(x-5): Symmetry & Intercepts

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Alright, guys, let's dive into graphing the function f(x) = (5x) / (x - 5). To get started, we need to figure out the symmetry of the graph. Symmetry helps us understand the behavior of the function and makes graphing much easier. We'll also pinpoint the y-intercept, which is another crucial point for sketching the graph accurately. So, grab your pencils and let's get to it!

Determining the Symmetry of the Graph

When we talk about symmetry, we're usually looking for two main types: y-axis symmetry and origin symmetry.

  • Y-axis symmetry means the graph looks the same on both sides of the y-axis. Mathematically, this happens when f(x) = f(-x) for all x in the domain. In simpler terms, if you plug in a positive x and a negative x of the same value, you get the same y value.
  • Origin symmetry means the graph is symmetric about the origin. This happens when f(-x) = -f(x) for all x in the domain. Imagine rotating the graph 180 degrees around the origin; if it looks the same, it has origin symmetry.

To determine the symmetry of f(x) = (5x) / (x - 5), we need to test these conditions.

Let's start by finding f(-x):

f(-x) = (5(-x)) / (-x - 5) = (-5x) / (-x - 5)

Now, let's see if f(x) = f(-x):

(5x) / (x - 5) = (-5x) / (-x - 5)

These two expressions are not equal, so the graph does not have y-axis symmetry.

Next, let's check if f(-x) = -f(x):

-f(x) = -(5x) / (x - 5) = (-5x) / (x - 5)

Now we compare f(-x) and -f(x):

(-5x) / (-x - 5) and (-5x) / (x - 5)

These two expressions are also not equal. Therefore, the graph does not have origin symmetry either.

So, the correct answer is: neither y-axis symmetry nor origin symmetry. This means the graph of the function doesn't have any of the typical symmetries we look for, which is quite common for rational functions.

Finding the Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. This occurs when x = 0. To find the y-intercept, we simply plug in x = 0 into the function f(x).

f(0) = (5(0)) / (0 - 5) = 0 / -5 = 0

So, the y-intercept is 0. This means the graph passes through the origin (0, 0).

Putting It All Together

We've determined that the graph of f(x) = (5x) / (x - 5) has neither y-axis symmetry nor origin symmetry, and its y-intercept is 0. This information is crucial for sketching the graph. Knowing that the graph isn't symmetric tells us we need to plot points on both sides of the y-axis to get an accurate picture. The y-intercept gives us one definite point on the graph: the origin.

To further sketch the graph, you might want to find the x-intercept (where f(x) = 0), look for any vertical or horizontal asymptotes, and plot a few more points. Remember, asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator of the function is zero, and horizontal asymptotes can be found by examining the behavior of the function as x approaches infinity.

In summary, analyzing the symmetry and finding intercepts are essential first steps in graphing any function. They provide a framework for understanding the function's behavior and allow for a more accurate sketch.

Asymptotes of the Function

Let's talk about asymptotes. Asymptotes are like guide rails for our graph, showing us where the function tends to as x gets really big (positive or negative) or as x approaches a value that makes the function undefined. For the function f(x) = (5x) / (x - 5), we want to find both vertical and horizontal asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero. This is because division by zero is undefined, and the function will approach infinity (or negative infinity) as x gets closer to that value. So, we need to find the values of x that make the denominator x - 5 equal to zero.

x - 5 = 0 x = 5

Therefore, we have a vertical asymptote at x = 5. This means as x approaches 5 from the left or the right, the function f(x) will shoot off towards positive or negative infinity. This is a critical piece of information for sketching our graph.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find horizontal asymptotes, we examine the degrees of the polynomials in the numerator and denominator. In our case, f(x) = (5x) / (x - 5), both the numerator and denominator are of degree 1 (the highest power of x is 1).

When the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 5, and the leading coefficient of the denominator is 1 (since x can be written as 1x). Therefore, the horizontal asymptote is:

y = 5 / 1 = 5

So, we have a horizontal asymptote at y = 5. This tells us that as x gets very large (positive or negative), the function f(x) will approach the value 5 but never actually reach it. Again, this is valuable information for sketching the graph.

Putting Asymptotes into Perspective

Knowing the asymptotes gives us a framework for understanding how the graph behaves at its extremes. The vertical asymptote at x = 5 tells us there's a break in the graph at that point, and the function will either shoot up to positive infinity or down to negative infinity as x approaches 5. The horizontal asymptote at y = 5 tells us the function will level out and approach 5 as x goes to positive or negative infinity.

When sketching the graph, you'll want to draw these asymptotes as dashed lines to guide your drawing. The graph will get closer and closer to these lines but never actually touch them (unless the graph crosses a horizontal asymptote at some point, which can happen).

In summary, for the function f(x) = (5x) / (x - 5), we've found:

  • Vertical asymptote: x = 5
  • Horizontal asymptote: y = 5

With this information, along with the symmetry analysis and the y-intercept we found earlier, we're well-equipped to sketch an accurate graph of the function.

Additional Points and Graphing Tips

Okay, so we've nailed down the symmetry (or lack thereof), the y-intercept, and the asymptotes. Now, let's talk about plotting some additional points and some general graphing tips to help you create a more accurate and detailed graph of f(x) = (5x) / (x - 5).

Plotting Additional Points

To get a better sense of the shape of the graph, it's always a good idea to plot a few extra points. Choose values of x that are on either side of the vertical asymptote (x = 5) and around the y-intercept (x = 0). Here are a few suggestions:

  • x = -2: f(-2) = (5(-2)) / (-2 - 5) = -10 / -7 = 10/7 ≈ 1.43
  • x = 2: f(2) = (5(2)) / (2 - 5) = 10 / -3 = -10/3 ≈ -3.33
  • x = 4: f(4) = (5(4)) / (4 - 5) = 20 / -1 = -20
  • x = 6: f(6) = (5(6)) / (6 - 5) = 30 / 1 = 30
  • x = 8: f(8) = (5(8)) / (8 - 5) = 40 / 3 ≈ 13.33

Plot these points on your graph. They'll give you a clearer picture of how the function behaves in different regions.

Graphing Tips

Here are some general tips to keep in mind as you sketch the graph:

  1. Start with the Asymptotes: Draw the vertical asymptote (x = 5) and the horizontal asymptote (y = 5) as dashed lines. These will guide your graph.

  2. Plot Key Points: Plot the y-intercept (0, 0) and any other points you've calculated.

  3. Connect the Dots: Carefully connect the points, making sure the graph approaches the asymptotes but doesn't cross them (unless it crosses the horizontal asymptote).

  4. Consider the Behavior Near Asymptotes: As x approaches the vertical asymptote (x = 5) from the left, the function will either go to positive or negative infinity. Use the points you've plotted to determine which direction it goes.

  5. Check for Intersections with the Horizontal Asymptote: A rational function can sometimes cross its horizontal asymptote. To find out if this happens, set f(x) equal to the value of the horizontal asymptote and solve for x.

    (5x) / (x - 5) = 5 5x = 5(x - 5) 5x = 5x - 25 0 = -25 (This is never true)

    Since there's no solution, the graph does not intersect the horizontal asymptote.

  6. Smooth Curves: Draw smooth curves, avoiding sharp corners or abrupt changes in direction.

  7. Label Everything: Label the axes, asymptotes, and any key points on your graph. This makes it easier to understand.

Common Mistakes to Avoid

  • Crossing Vertical Asymptotes: Remember, the graph should never cross a vertical asymptote.
  • Ignoring Asymptotes: Asymptotes are crucial for understanding the function's behavior, so make sure to use them as guides.
  • Plotting Too Few Points: The more points you plot, the more accurate your graph will be.
  • Assuming Symmetry: We already determined that this function doesn't have y-axis or origin symmetry, so don't assume it's symmetric.

By following these tips and avoiding common mistakes, you'll be well on your way to creating an accurate and informative graph of f(x) = (5x) / (x - 5). Happy graphing!