1. Identify a rational function whose graph is a horizontal line except for two holes. Graph the function.
2. Identify a rational function who graph lies entirely above the x-axis and has a single vertical asymptote. Graph the function.
3. Identify a rational function who graph has the horizontal asymptote y=2 and two vertical asymptotes. Graph the functions.
Could someone please explain to me how i would be able to figure out how to find these rational functions? And if it won't be any trouble to you, could you provide examples.. i really need to understand this because i have a quiz coming up. thanks in advance!
1. ((x-2)(x+3))/((x-2)(x+3))
Anything with reoccurring factors as the numerator and denominator will become a hole.
x^5÷x4-16
1. To find a rational function whose graph is a horizontal line except for two holes, we can start by considering a basic rational function of the form f(x) = c/(x - a), where c and a are constants. In this case, we want a horizontal line, so the numerator should be a constant.
Let's say we want the horizontal line to be at y = 3, and the two holes to be at x = -2 and x = 4. We can set up our rational function as:
f(x) = 3 / (x + 2)(x - 4)
To graph this function, we see that the numerator is constant, so the graph will be a horizontal line at y = 3. The denominator has two factors (x + 2) and (x - 4), so there will be two holes in the graph at x = -2 and x = 4.
2. To find a rational function whose graph lies entirely above the x-axis and has a single vertical asymptote, we can start by considering a basic rational function of the form f(x) = c/(x - a), where c and a are constants. In this case, we want the function to lie entirely above the x-axis, so the numerator should be positive.
Let's say we want the vertical asymptote to be at x = -3. We can set up our rational function as:
f(x) = 2 / (x + 3)
To graph this function, we see that the numerator is positive, so the graph will lie entirely above the x-axis. The denominator has a single factor (x + 3), so there will be a vertical asymptote at x = -3.
3. To find a rational function whose graph has the horizontal asymptote y = 2 and two vertical asymptotes, we can start by considering a basic rational function of the form f(x) = c/(x - a), where c and a are constants. In this case, we want the horizontal asymptote to be y = 2, so the numerator and denominator should have equal degrees.
Let's say we want the two vertical asymptotes to be at x = -1 and x = 2. We can set up our rational function as:
f(x) = (2x + 5) / ((x + 1)(x - 2))
To graph this function, we see that the denominator has two factors (x + 1) and (x - 2), so there will be two vertical asymptotes at x = -1 and x = 2. The numerator has a higher degree than the denominator, so the graph will not have a horizontal asymptote. However, we can find the behavior of the graph near positive and negative infinity to determine that there is a horizontal asymptote at y = 2.
To find rational functions that satisfy the given criteria, we need to understand the properties of rational functions and how they affect the graph. Here are the steps to find the functions:
1. For a rational function with a horizontal line graph except for two holes, we need to eliminate the holes by canceling out common factors in the numerator and denominator. Let's take the example of a horizontal line at y = 3 with holes at x = 1 and x = -1. The function would be:
f(x) = (x + 1)(x - 1) / (x - 1)(x + 1)
= (x + 1) / (x + 1)
= 1
The graph of this function is a horizontal line at y = 1, with holes at x = 1 and x = -1.
2. To find a rational function whose graph lies entirely above the x-axis and has a single vertical asymptote, we need to create a function that has a factor in the numerator that causes the denominator to approach zero at a particular value, resulting in a vertical asymptote.
Let's take the example of a vertical asymptote at x = 2. A function that satisfies this condition is:
f(x) = (x + 1)(x - 3) / (x - 2)(x - 2)
Notice that the denominator has a repeated factor (x - 2). This causes the rational function to have a vertical asymptote at x = 2. Graphing this function, you will see that it lies entirely above the x-axis.
3. To find a rational function with a horizontal asymptote at y = 2 and two vertical asymptotes, we need to create a function with factors that cause the numerator to approach zero at the vertical asymptotes and the denominator to become large for the horizontal asymptote.
Let's take the example of vertical asymptotes at x = -3 and x = 1, and a horizontal asymptote at y = 2. A function that satisfies these conditions is:
f(x) = (x + 3)(x - 1) / (x - 2)²
The numerator has factors that cause the rational function to approach zero at x = -3 and x = 1, resulting in vertical asymptotes. The denominator has a repeated factor that causes the function to approach y = 0. To shift this asymptote up to y = 2, we multiply the entire function by 2:
f(x) = 2(x + 3)(x - 1) / (x - 2)²
Graphing this function, you will see two vertical asymptotes and a horizontal asymptote at y = 2.
Remember, these are just examples to understand how to find rational functions with specific properties. You can modify the examples by incorporating additional factors or adjusting constants to create other rational functions with the desired characteristics.