Drawing Graphs of Functions
Guidelines for Graphing a Function
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.
How To: Draw the Graph of a Function
Given a function ff use the following steps to sketch a graph of ff:
- Step 1: Determine the domain of the function.
- Step 2: Locate the xx– and yy-intercepts.
- Step 3: Evaluate limx→∞f(x)limx→∞f(x) and limx→−∞f(x)limx→−∞f(x) to determine the end behavior.*
- Step 4: Determine whether ff has any vertical asymptotes.
- Step 5: Calculate f′f′. Find all critical points and determine the intervals where ff is increasing and where ff is decreasing. Determine whether ff has any local extrema.
- Step 6: Calculate f′′f′′. Determine the intervals where ff is concave up and where ff is concave down. Use this information to determine whether ff has any inflection points. The second derivative can also be used as an alternate means to determine or verify that ff has a local extremum at a critical point.
*Note for Step 3: If either of these limits is a finite number LL, then y=Ly=L is a horizontal asymptote. If either of these limits is ∞∞ or −∞−∞, determine whether ff has an oblique asymptote. If ff is a rational function such that f(x)=p(x)q(x)f(x)=p(x)q(x), where the degree of the numerator is greater than the degree of the denominator, then ff can be written as
where the degree of r(x)r(x) is less than the degree of q(x)q(x). The values of f(x)f(x) approach the values of g(x)g(x) as x→±∞x→±∞. If g(x)g(x) is a linear function, it is known as an oblique asymptote.
Now let’s use this strategy to graph several different functions. We start by graphing a polynomial function.
Sketch a graph of f(x)=(x−1)2(x+2)f(x)=(x−1)2(x+2)
Sketch the graph of f(x)=x21−x2
Sketch the graph of f(x)=x2x−1
Sketch a graph of f(x)=(x−1)23