using the 7 steps outlined in sec 4.3 of your book,analyze the graph of the following function R(x)=x^3-125/x^2-49
how do you expect us to know what section of 4.3 of your text is all about ??
Steve gave it a shot when a similar question was posted before, using a slightly different function, but the poster had actually inserted the " 7 steps ".
I suggest you look at his answer and apply it to your equation.
http://www.jiskha.com/display.cgi?id=1344386797
hint:
(x^3 - 125)/(x^2 - 49) factors,
= (x-5)(x^2 + 5x + 25)/( (x+7)(x-7) )
its the same steps just a different function in my textbook, but I don't understand it!
so, where do you get stuck? You must have some idea what each step entails. Rather than just do it all over again, I'd prefer to see what you get and what is confusing.
To analyze the graph of the function R(x) = (x^3 - 125) / (x^2 - 49) using the 7 steps outlined in section 4.3, here's how you can proceed:
Step 1: Find the domain
The denominator (x^2 - 49) cannot be zero, so we need to find the values of x that make it zero. Factoring the denominator, we get (x + 7)(x - 7). Therefore, the function is undefined at x = 7 and x = -7. The domain of R(x) is all real numbers except 7 and -7.
Step 2: Find the vertical asymptotes
Vertical asymptotes occur when the function approaches infinity as x approaches a certain value. To find the vertical asymptotes, we set the denominator equal to zero and solve for x. In this case, x^2 - 49 = 0 gives x = 7 and x = -7. These are the vertical asymptotes of the graph.
Step 3: Find the horizontal asymptotes
To find the horizontal asymptote, we examine the degrees of the numerator and denominator. The degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater, the function does not have a horizontal asymptote.
Step 4: Find the x-intercepts
The x-intercepts occur when the function equals zero. Setting R(x) = 0, we get x^3 - 125 = 0. By factoring the equation, we have (x - 5)(x^2 + 5x + 25) = 0. Thus, the x-intercept is x = 5.
Step 5: Find the y-intercept
The y-intercept occurs when x = 0. Plugging in x = 0 into the function, we get R(0) = (0^3 - 125)/(0^2 - 49) = (-125)/(-49) = 2.55 (rounded).
Step 6: Analyze the behavior around vertical asymptotes
We examine the behavior of the function as x approaches the vertical asymptotes (x = 7 and x = -7). We can do this by evaluating the function for values of x very close to these vertical asymptotes, both on the left and right side. For example, plugging in values such as 6.5, 6.9, -6.5, -6.9 into the function R(x) will give us an idea of how the function behaves close to the vertical asymptotes.
Step 7: Sketch the graph
Using all the information gathered from the previous steps, we can now sketch the graph of the function R(x). Plot the x-intercept, the y-intercept, the vertical asymptotes, and mark the behavior around the asymptotes. Additionally, you can find additional points by choosing different x-values and computing the corresponding y-values.
By following these 7 steps, you should be able to effectively analyze and sketch the graph of the given function R(x).