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.
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Part 1:
To graph a rational function with a linear binomial in both the numerator and denominator, we need to follow the steps below:
Step 1: Choose a linear binomial for the numerator and denominator. Let's say our linear binomial for the numerator is (2x + 3) and for the denominator is (x - 4).
Step 2: Write the rational function using the chosen linear binomials. The rational function will be:
f(x) = (2x + 3) / (x - 4)
Step 3: Use graphing technology (such as a graphing calculator or software) to graph the function. The graph will show the horizontal and vertical asymptotes, as well as the x- and y-intercepts.
The graphing technology will reveal that the vertical asymptote occurs at x = 4, the x-intercept is (-1.5, 0), and the y-intercept is (0, 0.75).
Part 2:
To determine the vertical asymptote, x-intercepts, and y-intercept for the rational function, we can do the following calculations:
1. Vertical Asymptote:
To find the vertical asymptote, we set the denominator of the rational function equal to zero and solve for x.
(x - 4) = 0
x = 4
So, the vertical asymptote occurs at x = 4.
2. X-Intercepts:
To find the x-intercepts, we set the numerator of the rational function equal to zero and solve for x.
2x + 3 = 0
2x = -3
x = -3/2
So, the x-intercept occurs at (-1.5, 0).
3. Y-Intercept:
To find the y-intercept, we substitute x = 0 into the rational function.
f(0) = (2(0) + 3) / (0 - 4)
f(0) = 3 / -4
f(0) = -0.75
So, the y-intercept occurs at (0, -0.75).
By following these steps, we have determined the vertical asymptote, the x-intercepts, and the y-intercept of the given rational function.