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.
what about the rest?
I don't understand
To create a rational function with a linear binomial in both the numerator and denominator, we will start by choosing linear binomials for both parts. Let's say we have (2x + 1) as the linear binomial in the numerator and (3x - 2) as the linear binomial in the denominator. Our rational function would then be:
f(x) = (2x + 1) / (3x - 2)
Now, let's move on to Part 1: graphing the function.
To graph the function, we can use graphing technology such as a graphing calculator or a graphing software. This will allow us to quickly visualize the function and determine its key features like horizontal and vertical asymptotes, and x- and y-intercepts.
Using a graphing calculator or software, follow these steps:
1. Enter the function f(x) = (2x + 1) / (3x - 2) into the calculator or software.
2. Set the window or viewing range to encompass the range of x-values you are interested in.
3. Plot the graph.
The resulting graph will show you the shape of the rational function and its key features: horizontal and vertical asymptotes, and x- and y-intercepts. Make sure to label these features on your graph.
Horizontal asymptotes:
To determine the horizontal asymptote(s) of a rational function, consider the degrees of the polynomials in the numerator and denominator. In this case, both the numerator and denominator are linear polynomials, which means the degrees are both 1.
When the degrees of the numerator and denominator are the same, as in this case, the horizontal asymptote can be found by dividing the leading coefficients. The leading coefficients here are 2 in the numerator and 3 in the denominator.
So, the equation of the horizontal asymptote is y = 2/3.
Vertical asymptotes:
To find the vertical asymptote(s) of a rational function, set the denominator equal to zero and solve for x. In this case, we set 3x - 2 = 0 and solve for x:
3x - 2 = 0
3x = 2
x = 2/3
Therefore, the vertical asymptote is x = 2/3.
X-intercepts:
To find the x-intercepts of a rational function, set the numerator equal to zero and solve for x. In this case, we set 2x + 1 = 0 and solve for x:
2x + 1 = 0
2x = -1
x = -1/2
Therefore, the x-intercept is x = -1/2.
Y-intercept:
To find the y-intercept of a rational function, substitute x = 0 into the function and solve for y. In this case, we substitute x = 0 into f(x) = (2x + 1) / (3x - 2):
f(0) = (2(0) + 1) / (3(0) - 2)
f(0) = 1 / (-2)
f(0) = -1/2
Therefore, the y-intercept is y = -1/2.
Now, you have all the information needed to graph the rational function and label the horizontal and vertical asymptotes, and the x- and y-intercepts.
how about
y = (3x - 4)/(x + 2)
Do you have a programmable calculator such as TI which does graphs?
How do you expect me to do this for you ?
There are all kinds of suitable webpages that will do graphs for you.
This one is one of the best:
http://www.wolframalpha.com/input/?i=plot+y+%3D+(3x+-+4)%2F(x+%2B+2)
The first graph is a close up, the second shows the whole thing within a reasonable domain