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 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.
To determine symmetry, we need to check if the function is even (symmetric about the y-axis) or odd (symmetric about the origin). In this case, we have a rational function, which is generally neither even nor odd. Therefore, it has no symmetry.
Step 2: Find the y-intercept.
To find the y-intercept, we set x = 0 and evaluate the function:
f(0) = (2(0)^2 + 0 - 3) / (2(0)^2 - 7(0))
= -3/0 (which is undefined)
Since the denominator is zero, there is no y-intercept for this function.
Step 3: Find the x-intercepts.
To find the x-intercepts, we set f(x) = 0 and solve for x:
2x^2 + x - 3 = 0
This quadratic equation can be factored or solved using the quadratic formula.
After factoring or using the quadratic formula, we obtain the x-intercepts as x = -3/2 and x = 1/2.
Step 4: Find the vertical asymptotes.
To find the vertical asymptotes, we determine the values of x that make the denominator equal to zero (i.e., the values that lead to division by zero):
2x^2 - 7x = 0
This equation can be factored to find the values of x that lead to division by zero.
The vertical asymptotes occur when x = 0 and x = 7/2.
Step 5: Find the horizontal asymptotes.
To find the horizontal asymptotes, we investigate the behavior of the function as x approaches positive and negative infinity.
For rational functions, the degrees of the numerator and denominator determine the horizontal asymptotes. Since the degree of the numerator (2) is less than the degree of the denominator (2), there will be a horizontal asymptote at y = 0.
Step 6: Plot points between and beyond the x-intercepts and the vertical asymptotes.
To plot points, we can choose values of x between and beyond the x-intercepts and vertical asymptotes.
For example, let's choose the following values of x: -5, -2, -1, 2, and 5.
- For x = -5:
f(-5) = (2(-5)^2 + (-5) - 3) / (2(-5)^2 - 7(-5))
= -27/19
- For x = -2:
f(-2) = (2(-2)^2 + (-2) - 3) / (2(-2)^2 - 7(-2))
= -5/17
- For x = -1:
f(-1) = (2(-1)^2 + (-1) - 3) / (2(-1)^2 - 7(-1))
= 0
- For x = 2:
f(2) = (2(2)^2 + (2) - 3) / (2(2)^2 - 7(2))
= -15/4
- For x = 5:
f(5) = (2(5)^2 + (5) - 3) / (2(5)^2 - 7(5))
= -13/69
Step 7: Graph the points and join them to create the graph of the function.
Based on the information provided, the graph of the rational function should have the x-intercepts at x = -3/2 and x = 1/2, vertical asymptotes at x = 0 and x = 7/2, a horizontal asymptote at y = 0, and passes through the points (-5, -27/19), (-2, -5/17), (-1, 0), (2, -15/4), and (5, -13/69).