Draw the graph of the function (qualitatively accurate).
y=x+1−−−−−√/(x−b),1<b<2
To draw the graph of the function y = (x + 1 )^1/2 / (x - b) for the given conditions, we can follow these steps:
1. Determine the domain of the function:
- In this case, the only restriction is that b should be between 1 and 2, given as 1 < b < 2. So, the domain of the function is all real numbers except for when x = b.
2. Find the x-intercept(s):
- To find the x-intercept(s), we set y = 0 and solve for x.
- In this case, since y is a square root function, it can never be zero because the square root of any positive number is always positive. So, there are no x-intercepts.
3. Determine vertical asymptote(s):
- Vertical asymptotes occur when the denominator of a rational function becomes zero.
- In this case, the denominator is (x - b), so the vertical asymptote occurs at x = b.
4. Determine horizontal asymptote(s):
- To determine the horizontal asymptote(s), we need to compare the degrees of the numerator and denominator.
- In this case, the numerator has a degree of 1 (from the square root) and the denominator has a degree of 1. Therefore, there is no horizontal asymptote, and the function approaches infinity or negative infinity as x approaches infinity or negative infinity, respectively.
5. Find the y-intercept:
- To find the y-intercept, we set x = 0 and solve for y.
- Plugging in x = 0, we have y = (0 + 1 )^1/2 / (0 - b) = 1^1/2 / -b = 1 / -b. So, the y-intercept is (0, 1 / -b).
6. Plot additional points:
- Choose a few x-values within the domain of the function and calculate the corresponding y-values to plot additional points on the graph.
- For example, you can choose x = -3, -2, -1, 1, 2, and 3, and find the corresponding y-values using the formula.
7. Sketch the graph:
- Using the points obtained from the calculations and considering the asymptotes, sketch the graph of the function in the coordinate plane.
Remember to label the important points (such as intercepts and asymptotes) and the axes to create a clear and accurate graph of the function.