Rational Function: Find A, K, And Graph Y = A/(x-7) + K
Hey guys! Today, we're diving deep into the world of rational functions. Specifically, we're going to tackle a problem involving a rational function of the form y = a/(x-7) + k. This type of function has some interesting properties, and understanding how to work with it is crucial for many areas of math and even real-world applications. We’ll figure out how to find the values of 'a' and 'k' given some points, and then we'll graph the function. So, buckle up and let's get started!
Understanding Rational Functions
Before we jump into the problem, let's quickly recap what rational functions are all about. A rational function is essentially a function that can be expressed as a ratio of two polynomials. In our case, we have y = a/(x-7) + k. The denominator x - 7 is what makes this a rational function. The presence of a variable in the denominator introduces the possibility of vertical asymptotes, which are vertical lines that the function approaches but never actually touches. These asymptotes play a crucial role in the shape and behavior of the graph.
Asymptotes are like invisible barriers that guide the function's path. A vertical asymptote occurs where the denominator of the rational function equals zero. In our case, the denominator is x - 7, so the vertical asymptote is at x = 7. This means the function will get incredibly close to the line x = 7 but will never cross it. There's also a horizontal asymptote, which in this form of the function, is simply y = k. The value of k represents a vertical shift of the basic rational function y = 1/x. Understanding these asymptotes is essential for accurately graphing the function.
Rational functions are powerful tools for modeling real-world situations where quantities are inversely proportional. For example, they can be used to describe the relationship between the rate of travel and the time it takes to cover a certain distance, or the relationship between the number of workers and the time it takes to complete a task. They also appear in fields like physics, engineering, and economics. By mastering rational functions, you're not just learning math; you're gaining a skillset that can be applied in various practical contexts. So, let's move on to the problem and see how we can put these concepts into action.
Determining the Values of 'a' and 'k'
Our main task here is to find the values of 'a' and 'k' in the rational function y = a/(x-7) + k. We're given two points that the function passes through: (10, 1) and (2, 9). This is our key to unlocking the solution! Since these points lie on the graph of the function, their coordinates must satisfy the equation. In other words, if we plug in the x-value of a point into the equation, we should get the corresponding y-value. We can use this information to create a system of two equations with two unknowns ('a' and 'k'), which we can then solve.
Let's start with the point (10, 1). Substituting x = 10 and y = 1 into the equation y = a/(x-7) + k, we get:
1 = a/(10-7) + k
Simplifying this, we have:
1 = a/3 + k
This is our first equation. Now, let's do the same with the point (2, 9). Substituting x = 2 and y = 9 into the equation, we get:
9 = a/(2-7) + k
Simplifying, we have:
9 = a/(-5) + k
Or, more neatly:
9 = -a/5 + k
This is our second equation. Now we have a system of two linear equations:
- 1 = a/3 + k
- 9 = -a/5 + k
We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method. To do this, we'll subtract the first equation from the second equation. This will eliminate k, leaving us with an equation in terms of a only. Subtracting equation 1 from equation 2, we get:
9 - 1 = (-a/5 + k) - (a/3 + k)
8 = -a/5 - a/3
Now, let's find a common denominator for the fractions, which is 15. We get:
8 = -3a/15 - 5a/15
Combining the terms with a, we have:
8 = -8a/15
To solve for a, we multiply both sides by -15/8:
a = 8 * (-15/8)
a = -15
So, we've found that a = -15. Now we can substitute this value back into either equation 1 or equation 2 to solve for k. Let's use equation 1:
1 = a/3 + k
Substitute a = -15:
1 = -15/3 + k
1 = -5 + k
Adding 5 to both sides, we get:
k = 6
Therefore, we've determined that a = -15 and k = 6. Awesome! We've successfully found the values of 'a' and 'k' for our rational function. Now, let's move on to graphing the function.
Graphing the Function
Now that we know the values of a and k, we can write the specific equation of our rational function: y = -15/(x-7) + 6. To graph this function, we'll use the information we've gathered so far, including the values of a and k, the asymptotes, and the points we were given.
First, let's identify the asymptotes. We already know that the vertical asymptote is at x = 7, because that's where the denominator x - 7 equals zero. The horizontal asymptote is y = k, and since we found that k = 6, the horizontal asymptote is y = 6. These asymptotes will serve as our guidelines for sketching the graph.
Next, let's plot the points we were given: (10, 1) and (2, 9). These points will help us determine the shape of the graph in each region separated by the vertical asymptote. Remember, rational functions have distinct behaviors on either side of their vertical asymptotes. They can either approach positive or negative infinity as x gets closer to the asymptote.
Now, let's analyze the effect of the value of a. In our case, a = -15, which is negative. This means that the graph will be reflected across the x-axis compared to the basic rational function y = 1/x. Specifically, on the right side of the vertical asymptote (x > 7), the function will approach the horizontal asymptote from below, and on the left side of the vertical asymptote (x < 7), the function will approach the horizontal asymptote from above.
With all this information in hand, we can now sketch the graph. Draw the vertical asymptote at x = 7 and the horizontal asymptote at y = 6. Plot the points (10, 1) and (2, 9). On the right side of the vertical asymptote, the graph will pass through (10, 1) and approach the horizontal asymptote y = 6 from below. On the left side of the vertical asymptote, the graph will pass through (2, 9) and approach the horizontal asymptote y = 6 from above. The graph will also approach the vertical asymptote x = 7 without ever touching it. The negative value of 'a' causes the graph to be in the second and fourth quadrants relative to the intersection of the asymptotes.
The graph will consist of two separate curves, one on each side of the vertical asymptote. Each curve will smoothly approach both the vertical and horizontal asymptotes. By combining our knowledge of the asymptotes, the plotted points, and the effect of the value of a, we can create an accurate representation of the rational function y = -15/(x-7) + 6.
Conclusion
Alright, guys, we did it! We successfully found the values of 'a' and 'k' for the given rational function and then graphed it. We saw how using the given points allowed us to create a system of equations, which we then solved using the elimination method. We also learned how to identify and use asymptotes to guide our graphing process. Understanding the effect of the value of 'a' on the graph's shape was also crucial. By breaking down the problem step by step, we were able to tackle this complex function with confidence.
Working with rational functions like this is not just about memorizing formulas and procedures. It's about developing a deep understanding of how functions behave and how different parameters affect their graphs. These skills are invaluable in many areas of mathematics, science, and engineering. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries! You've got this! Remember, the world of mathematics is vast and fascinating, and there's always something new to discover. Keep up the great work, and I'll see you in the next math adventure!