Create a rational function with a linear binomial in both the numerator and denominator.

Part 1. Graph your function using technology. Include the horizontal and vertical asymptotes and the x- and y-intercepts on your graph. Label the asymptotes and intercepts.
Part 2. Show all work to identify the vertical asymptote, the x-intercepts, and the y-intercept.

y = (2x-3)/(7x+1)

see

http://www.wolframalpha.com/input/?i=%282x-3%29%2F%287x%2B1%29

thank you, steve.

Part 1:

To graph the rational function with a linear binomial in both the numerator and denominator, let's consider the following function:

f(x) = (2x + 3) / (5x - 1)

Using technology, we can graph this function by plugging it into a graphing calculator or a graphing software. Here is a graph of the function:

[Graph of f(x) = (2x + 3) / (5x - 1)]

In this graph, we can observe the following:

- Vertical asymptote: The vertical asymptote occurs when the denominator of the rational function equals zero, so we have 5x - 1 = 0. Solving this equation, we find x = 1/5. Therefore, the vertical asymptote is x = 1/5.

- X-intercepts: The x-intercepts occur when the numerator of the rational function equals zero, so we have 2x + 3 = 0. Solving this equation, we find x = -3/2. Therefore, the x-intercept is x = -3/2.

- Y-intercept: The y-intercept occurs when x = 0, so we can plug in x = 0 into the function to find the value of the y-intercept. Evaluating f(0), we get f(0) = (2(0) + 3) / (5(0) - 1) = 3 / (-1) = -3. Therefore, the y-intercept is y = -3.

Part 2:
To find the vertical asymptote, x-intercepts, and y-intercept algebraically, we follow these steps:

- Vertical asymptote: Set the denominator equal to zero and solve for x. In our function, the denominator is 5x - 1. So, we have 5x - 1 = 0. Solving for x, we get x = 1/5. Therefore, the vertical asymptote is x = 1/5.

- X-intercepts: Set the numerator equal to zero and solve for x. In our function, the numerator is 2x + 3. So, we have 2x + 3 = 0. Solving for x, we get x = -3/2. Therefore, the x-intercept is x = -3/2.

- Y-intercept: Set x = 0 in the function and evaluate. In our function, we have f(0) = (2(0) + 3) / (5(0) - 1) = 3 / (-1) = -3. Therefore, the y-intercept is y = -3.

Overall, the rational function f(x) = (2x + 3) / (5x - 1) has a vertical asymptote at x = 1/5, an x-intercept at x = -3/2, and a y-intercept at y = -3.

To create a rational function with a linear binomial in both the numerator and denominator, we can start by choosing two linear functions, one for the numerator and one for the denominator. Let's use the following examples:

Numerator: f(x) = 3x + 2
Denominator: g(x) = 2x - 1

Part 1: Graphing the function using technology

To graph the rational function and determine the asymptotes and intercepts, we can use a graphing calculator or online graphing tool. Here's how to do it using an online graphing tool:

Step 1: Open a graphing tool like Desmos (https://www.desmos.com/calculator).

Step 2: Enter the expression for the rational function as follows:

y = (3x + 2) / (2x - 1)

Step 3: Click on the graph button to see the graph.

Step 4: Analyze the graph to identify the asymptotes and intercepts.

- Horizontal asymptote: Look at the end behavior of the function. As x approaches positive or negative infinity, the numerator's degree (1) is less than the denominator's degree (1), resulting in a horizontal asymptote at the ratio of the leading coefficients, which is 3 / 2. Label this horizontal asymptote on the graph.

- Vertical asymptote: The vertical asymptote occurs when the denominator is equal to zero. In this case, set the denominator (2x - 1) equal to zero and solve for x to find the vertical asymptote. In this case, x = 1/2 is the vertical asymptote. Label this vertical asymptote on the graph.

- X-intercepts: The x-intercepts occur when the numerator is equal to zero. In this case, set the numerator (3x + 2) equal to zero and solve for x. The x-intercept is x = -2/3. Label this x-intercept on the graph.

- Y-intercept: The y-intercept occurs when x is equal to zero. Substitute x = 0 into the function to find the y-intercept. In this case, the y-intercept is y = 2/(-1) = -2. Label this y-intercept on the graph.

Part 2: Identifying the asymptotes and intercepts

To find the vertical asymptote, x-intercepts, and y-intercept algebraically, we can do the following:

Vertical asymptote: Set the denominator (2x - 1) equal to zero and solve for x:

2x - 1 = 0
2x = 1
x = 1/2

So the vertical asymptote is x = 1/2.

X-intercepts: Set the numerator (3x + 2) equal to zero and solve for x:

3x + 2 = 0
3x = -2
x = -2/3

So the x-intercept is x = -2/3.

Y-intercept: Substitute x = 0 into the function:

y = (3(0) + 2) / (2(0) - 1)
y = 2 / -1
y = -2

So the y-intercept is y = -2.

By graphing the function and performing the algebraic steps, we have determined the vertical asymptote, x-intercepts, and y-intercept of the rational function.