(Using the seven steps to analyzing the rational function of R), analyze the graph of the following function.
R(x)=x^2-49/x^4-16
To analyze the graph of the given rational function, R(x) = (x^2 - 49)/(x^4 - 16), we can follow the seven steps to analyze rational functions:
Step 1: Identify the domain
The domain of a rational function is all the real numbers except for the values that would make the denominator zero. In this case, the denominator x^4 - 16 will be zero if x^4 - 16 = 0. Solving this equation, we get x = ±2. Therefore, the domain of R(x) is all real numbers except x = ±2.
Step 2: Find the vertical asymptotes
Vertical asymptotes occur at values of x where the rational function is undefined, i.e., when the denominator is zero. In this case, the vertical asymptotes occur at x = ±2, which are the values excluded from the domain.
Step 3: Find the horizontal asymptotes
To determine horizontal asymptotes, we need to compare the degrees of the numerator and the denominator functions. In this case, the degree of the numerator is 2 and the degree of the denominator is 4. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote will be y = 0, which is the x-axis.
Step 4: Find the x-intercepts
X-intercepts occur when the numerator of the rational function is zero. Therefore, to find them, we solve the equation x^2 - 49 = 0. Simplifying, we get x = ±7. So the x-intercepts are x = ±7.
Step 5: Find the y-intercept
The y-intercept occurs when x = 0. Evaluating the function at x = 0, we get R(0) = (0^2 - 49)/(0^4 - 16) = -49/(-16) = 49/16. Therefore, the y-intercept is at y = 49/16.
Step 6: Determine the symmetry
To check for symmetry in a rational function, we need to examine if the function is even, odd, or neither. In this case, R(x) is neither even nor odd since the powers in the numerator and the denominator are not the same.
Step 7: Sketch the graph
Using the information from the previous steps, we can sketch the graph of the rational function. The graph will have vertical asymptotes at x = ±2, a horizontal asymptote at y = 0, x-intercepts at x = ±7, a y-intercept at y = 49/16, and no symmetry.
I hope this explanation helps in understanding how to analyze the graph of a rational function using the seven steps!