Use the seven step method described in the book to graph the following rational function
f(x)=(2x^2+x-3)/(2x^2-7x)
1) Determine the symmetry of the function
2) Find the y-intercept
3) Find the x-intercept
4) Find the vertical asymptotes
5) Find the horizontal asymptotes
6) Plot points between and beyond the x-intercept and the vertical asymptotes.
Type the y-coordinate for each of the following points:
-5 -2 -1 2 5
7) What should the graph look like
To graph the given rational function f(x) = (2x^2 + x - 3)/(2x^2 - 7x), we can follow the seven-step method:
Step 1: Determine the symmetry of the function
A rational function is symmetric about the y-axis if it satisfies f(-x) = f(x). To check this, substitute -x in place of x in the given function and simplify:
f(-x) = (2(-x)^2 + (-x) - 3)/(2(-x)^2 - 7(-x))
= (2x^2 - x - 3)/(2x^2 + 7x)
Since f(-x) ≠ f(x), the function is not symmetric about the y-axis.
Step 2: Find the y-intercept
To find the y-intercept, substitute x = 0 into the function:
f(0) = (2(0)^2 + 0 - 3)/(2(0)^2 - 7(0)) = (-3)/(0)
The denominator is zero, indicating that there is a vertical asymptote at x = 0. Therefore, there is no y-intercept.
Step 3: Find the x-intercept
To find the x-intercept, let y = 0 and solve for x:
0 = (2x^2 + x - 3)/(2x^2 - 7x)
Cross-multiply to get:
0 = 2x^2 + x - 3
Now, factor the quadratic equation (if possible) or use the quadratic formula to find the x-intercepts.
Step 4: Find the vertical asymptotes
To find the vertical asymptotes, set the denominator equal to zero and solve for x:
2x^2 - 7x = 0
Factor out x:
x(2x - 7) = 0
Set each factor equal to zero:
x = 0 or 2x - 7 = 0
x = 0 or x = 7/2
So, there are two vertical asymptotes: x = 0 and x = 7/2.
Step 5: Find the horizontal asymptotes
To find the horizontal asymptote(s), we compare the degrees of the numerator and denominator.
Since the degree of the numerator is 2 (highest power of x) and the degree of the denominator is also 2, we compare the coefficients of the highest power.
Coefficients:
Numerator: 2
Denominator: 2
The ratio of the coefficients is 2/2 = 1.
Therefore, the horizontal asymptote is at y = 1.
Step 6: Plot points between and beyond the x-intercept and the vertical asymptotes
Plotting the points between and beyond the x-intercept and the vertical asymptotes will help us visualize the shape of the graph.
Let's evaluate the function at x = -5, -2, -1, 2, and 5 to get the corresponding y-values:
For x = -5:
f(-5) = (2(-5)^2 + (-5) - 3)/(2(-5)^2 - 7(-5)) = -17/67
For x = -2:
f(-2) = (2(-2)^2 + (-2) - 3)/(2(-2)^2 - 7(-2)) = 11/10
For x = -1:
f(-1) = (2(-1)^2 + (-1) - 3)/(2(-1)^2 - 7(-1)) = -2/3
For x = 2:
f(2) = (2(2)^2 + (2) - 3)/(2(2)^2 - 7(2)) = -9/5
For x = 5:
f(5) = (2(5)^2 + (5) - 3)/(2(5)^2 - 7(5)) = -10/33
Step 7: What should the graph look like
Based on the above information, the graph of the rational function f(x) should have the following characteristics:
- It is not symmetric about the y-axis.
- There is no y-intercept.
- The x-intercepts need to be found by solving the quadratic equation from Step 3.
- There are two vertical asymptotes located at x = 0 and x = 7/2.
- The horizontal asymptote is at y = 1.
- Plot the points (-5, -17/67), (-2, 11/10), (-1, -2/3), (2, -9/5), and (5, -10/33).
- The graph should show the behavior of approaching the vertical asymptotes and the horizontal asymptote as x approaches positive and negative infinity.
By following these steps, you should be able to graph the given rational function f(x).
f(x) = (2x+3)(x-1) / x(2x-7)
So, the denominator is zero at x=0,7/2 making vertical asymptotes
Top and bottom are both quadratics with x^2 coefficient=2, so there is a horizontal asymptote at y = 2/2 = 1
There is no y-intercept, because f(x) is not defined for x=0
It's easy to see where y=0
f(x) is neither even nor odd, so there is no symmetry
visit wolframalpha.com to check out the graph. Just type in the function