use the seven step method to graph the following rational function:
f(x)=3x^2+x-4/2x^2-5x
a. determine the symmetry of the function.
b. find the x and y intercepts.
c. find the verticle asymptote
d.find the horizontal asymptote
e.plot the points between and beyond the x intercept and verticle asymptote.
I don't get any of this. help me please?
How can you not get any of this?
symmetry may be a bit tough:
f(x) = (3x+4)(x-1) / x(2x-5)
I don't see any symmetry there at all.
the intercepts are easy, right?
Vertical asymptote where denominator si zero
horizontal asymptote is y when x gets huge. There, all the small powers of x become insignificant, and you are working with
3x^2/2x^2 = 3/2
visit wolframalpha.com and enter
(3x^2+x-4)/(2x^2-5x)
for a graph
Sure! I'll walk you through the seven-step method to graphing the given rational function.
Step 1: Determine the Symmetry of the Function
To determine the symmetry of the function, we need to check if it is an even or odd function. For a rational function, the general rule is that if the function has no negative exponents, it is an even function, meaning it is symmetrical about the y-axis. In our case, the function has no negative exponents, so it is an even function. Therefore, it is symmetric about the y-axis.
Step 2: Find the x and y-intercepts
To find the x-intercepts, set the numerator equal to zero and solve for x:
3x^2 + x - 4 = 0
You can use factoring or the quadratic formula to solve this equation. After finding the values of x, these are the x-intercepts.
To find the y-intercept, plug in x=0 into the function:
f(0) = (3(0)^2 + 0 - 4) / (2(0)^2 - 5(0))
Simplify the expression and solve. The resulting value is the y-intercept.
Step 3: Find the Vertical Asymptote
To find the vertical asymptote, set the denominator equal to zero and solve for x:
2x^2 - 5x = 0
Factor out x and solve for x=0, x=5/2.
These are the vertical asymptotes.
Step 4: Find the Horizontal Asymptote
To find the horizontal asymptote, we look at the degrees of the numerator and the denominator. Since the degree of the numerator is equal to 2 and the degree of the denominator is also equal to 2, we divide the leading coefficients of both the numerator and denominator to find the horizontal asymptote.
Step 5: Plot Points between and Beyond x-intercepts and Vertical Asymptote
Choose a few x-values between and beyond the x-intercepts and vertical asymptote. Plug them into the function to find the corresponding y-values. Plot these points on the graph.
Step 6: Graph the x and y-intercepts
Plot the x-intercepts and the y-intercept on the graph.
Step 7: Sketch the Rational Function
Using the information obtained from the previous steps, sketch the graph of the rational function, paying attention to the symmetry, asymptotes, and plotted points.
Note: Since it is difficult to graph here, I recommend using graphing software or a graphing calculator to plot the points and sketch the function accurately.
I hope this helps you understand the seven-step method for graphing a rational function! Let me know if you have any more questions.