using the 8 steps in Sec 4.3 of your book, analyze the graph of the following function.
R(x)= x^2+x-30/ x^2-x-20
Without knowing the 8 steps to which you are referring, you can do the following:
Factor both numerator and denominator:
R(x)=(x-5)(x+6)/[(x-5)(x+4)]
You will find the zeroes and asymptotes.
Since there is a common factor (x-5), the point f(5) is indeterminate and should be taken out of the domain.
Check against your eight steps.
If it's not clear, please post which step you need further explanations.
To analyze the graph of the function R(x) = (x^2 + x - 30)/(x^2 - x - 20) using the 8 steps in Section 4.3 of your book, you can follow these steps:
Step 1: Find the domain of the function R(x).
To determine the domain, we need to find the values of x for which the function is defined. In this case, since the denominator (x^2 - x - 20) cannot be equal to zero, we solve the equation x^2 - x - 20 = 0. The solutions to this quadratic equation are x = -4 and x = 5. However, we need to exclude these values from the domain since they make the function undefined. Therefore, the domain of R(x) is all real numbers except x = -4 and x = 5.
Step 2: Find the vertical asymptotes.
Vertical asymptotes occur when the denominator of a rational function is equal to zero. In this case, the equation x^2 - x - 20 = 0 gives us the values x = -4 and x = 5. Therefore, the vertical asymptotes for R(x) are x = -4 and x = 5.
Step 3: Find the horizontal or slant asymptotes (if any).
To determine horizontal or slant asymptotes, we compare the degrees of the numerator and denominator polynomials. If the degree of the numerator is equal to or smaller than the denominator, there is a horizontal asymptote at y = 0. However, if the degree of the numerator is larger than the denominator by exactly 1, there is a slant asymptote. In this case, the degree of the numerator and denominator polynomials is both 2. Therefore, there are no horizontal or slant asymptotes for R(x).
Step 4: Find the x-intercepts (if any).
X-intercepts occur when the value of y is zero. To find the x-intercepts of R(x), we set the numerator to zero and solve the equation x^2 + x - 30 = 0. Factoring or using the quadratic formula, we find x = -6 and x = 5 as the x-intercepts.
Step 5: Find the y-intercept (if any).
The y-intercept occurs when x is equal to zero. Substituting x = 0 into the function R(x), we get R(0) = (0^2 + 0 - 30)/(0^2 - 0 - 20) = -30/-20 = 3/2. Therefore, the y-intercept is (0, 3/2).
Step 6: Determine the symmetry (if any).
To determine the symmetry of the graph of R(x), we check if R(x) = R(-x). If R(x) is equal to R(-x) for all values of x in the domain, the function is even. If R(-x) equals the additive inverse of R(x), the function is odd. For this particular function, R(x) is neither even nor odd.
Step 7: Sketch the graph based on the previous steps.
Using the information obtained from the previous steps, you can now sketch the graph of R(x) on a coordinate plane. Plot the vertical asymptotes, x-intercepts, and y-intercept, and check the behavior of the function as x approaches positive and negative infinity to determine the end behavior.
Step 8: Test additional points (optional).
To further confirm your graph and gain a clearer understanding of the behavior of the function, you can choose some additional test points in the domain and evaluate them in the function. Plot these points on the graph to ensure they align with your graph's shape and trends.
By following these steps, you can analyze and sketch the graph of the given function R(x).