• Does the graph have any symmetries? Briefly consider if the function is even \(\bigl((f(x) = f(-x)\bigr)\) or odd \(\bigl(f(x) = -f(-x)\bigr)\) or if the function is periodic like the trigonometric functions sine, cosine, secant, etc
• Where are the graph’s intercepts? The \(y\)-intercept is \(\bigl(0, f(0)\bigr).\) The \(x\)-intercepts are all points \(\bigl(c, 0\bigr)\) at which \(f(c) = 0.\) If \(f(x)\) is a fractional formula, it’s sufficient to calculate the zeros of the numerator.
• Where are the graph’s vertical asymptotes? I.e. where are the function’s non-removable discontinuities? If \(f(x)\) is a fractional formula, the roots of the denominator are called the poles of \(f\); vertical asymptotes occur at the poles.
• Where are the graph’s horizontal asymptotes? I.e. what’s the function’s “end-behaviour”? Evaluate the limits \(\lim_{x \to \pm \infty} f(x).\) If \(f\) is a rational function, what is the quotient of the leading terms of its numerator and denominator?
• Where is the graph increasing or decreasing? Compute \(f'\) and determine the \(c\) in the domain of \(f\) for which either \(f'(c) = 0\) or \(f'(c)\) is not defined; these are critical numbers. Within each interval between adjacent pairs of these critical numbers test a single value of \(x:\) if \(f'(x) \gt 0\) then the graph will be increasing on that interval, and if \(f'(x) \lt 0\) then the graph will be decreasing on that interval.
• Where is the graph concave up or concave down? Compute \(f''\) and determine the \(c\) in the domain of \(f\) for which either \(f''(c) = 0\) or \(f''(c)\) is not defined; these are potential inflection locations. Within each interval between adjacent pairs of these potential inflection locations test a single value of \(x:\) if \(f''(x) \gt 0\) then the graph will be concave up on that interval, and if \(f''(x) \lt 0\) then the graph will be concave down on that interval.