Sketch the graph of f(x)=x+2/x-4. Find the vertical and horizontal asymptotes, critical points, concave up/down, and inflection points.
To sketch the graph of the given function, f(x) = (x + 2) / (x - 4), we can follow these steps:
1. Identify the vertical asymptote(s):
To find the vertical asymptote(s), we set the denominator equal to zero and solve for x:
x - 4 = 0
x = 4
Therefore, the vertical asymptote is x = 4.
2. Determine the horizontal asymptote:
To find the horizontal asymptote, we look at the degrees of the numerator and denominator.
The degree of the numerator is 1 (x) and the degree of the denominator is also 1 (x).
In this case, when the degrees are the same, the horizontal asymptote is determined by the ratio of the leading coefficients.
The ratio of the leading coefficients here is 1 / 1 = 1.
Thus, the horizontal asymptote is y = 1.
3. Locate the critical point(s):
To find the critical point(s), we set the derivative of the function equal to zero.
Let's find the derivative of f(x):
f'(x) = (x - 4) - (x + 2) / (x - 4)^2
Simplifying further, we have:
f'(x) = (x - 4 - x - 2) / (x - 4)^2
f'(x) = -6 / (x - 4)^2
Setting f'(x) = 0, we find that there are no critical points, as the numerator is a constant.
4. Determine concave up/down:
To determine the concavity, we need the second derivative.
Let's find the second derivative, f''(x):
f''(x) = -6(-2) / (x - 4)^3
f''(x) = 12 / (x - 4)^3
Since the second derivative is positive for all x, the function is concave up everywhere.
5. Identify inflection point(s):
To find the inflection point(s), we set the second derivative equal to zero and solve for x.
However, in this case, the second derivative is never equal to zero.
Therefore, there are no inflection points.
Now let's plot the graph using the information we've gathered:
- There is a vertical asymptote at x = 4.
- There is a horizontal asymptote at y = 1.
- The graph is concave up everywhere.
- There are no critical points or inflection points.
The graph should resemble a hyperbola, with vertical and horizontal asymptotes, without any specific points of interest.