Section 4.6 Sketching Graphs of Functions
Subsection 4.6.1 Curve Sketching
Consider a function \(f(x)\text{.}\) The function itself, and its first and second derivative, provide useful information about the graph of \(y = f(x)\text{.}\) From pre-calculus, we can determine,
- Points on the graph of \(y = f(x)\text{,}\) by evaluating \(f\text{.}\)
- The domain and range of \(f\text{.}\)
- The \(x\)-intercepts and \(y\)-intercepts.
- Symmetry, if the function is even or odd.
Using calculus, we can determine additional properties of a graph.
- Using limits, we can determine continuity (in particular, points of discontinuity), and any vertical or horizontal asymptotes.
- From the first derivative \(f'\text{,}\) we can determine intervals of increase and decrease, and any local extrema.
- From the second derivative \(f''\text{,}\) we can determine concavity and inflection points.
Notice the importance of algebra for solving the equations \(f(x) = 0\) (to determine the \(x\)-intercepts), \(f'(x) = 0\) (to determine the critical points), and \(f''(x) = 0\) (to determine the inflection points). In summary, calculus allows us to sketch graphs of functions more accurately.
Subsection 4.6.2 Sketching the Graph of a Polynomial Function
First, we will consider sketching the graph of a polynomial function \(y = f(x)\text{.}\) The following steps provide a good outline as to how to sketch the graph of \(y = f(x)\text{.}\)
- Intercepts. Determine any \(x\)-intercepts and/or the \(y\)-intercept, by solving the polynomial equation \(f(x) = 0\text{,}\) and by evaluating \(y = f(0)\text{,}\) respectively.
- Symmetry. Determine any symmetry, i.e. if \(f\) is an even or odd function.
- Extrema and intervals of increase/decrease. Determine \(f'(x)\text{,}\) and determine any critical points by solving \(f'(x) = 0\text{,}\) and determine any local extrema using the first or second derivative test. Determine the intervals where \(f\) is increasing and decreasing.
- Concavity and inflection points. Determine \(f''(x)\text{,}\) and determine any potential inflection points by solving \(f''(x) = 0\text{.}\) Determine the intervals where \(f\) is concave up or down, and use this to determine the actual inflection points.
- Plot the intercepts, critical points, and inflection points. Recall that critical points indicate horizontal tangents, and inflection points indicate changing concavity.
- Polynomial end behavior. Consider the end behavior of \(f\text{,}\) i.e. the behavior of \(f\) as \(x \to \infty\) and \(x \to -\infty\text{.}\)
- Use all preceding information to determine the overall shape of the graph. If necessary, plot additional points.
- Finally, connect the points with a smooth curve.
Subsection 4.6.3 Quadratic Functions (Parabolas)
Consider a quadratic function \(f(x) = ax^2 + bx + c\text{.}\) You might recall from pre-calculus that the vertex of this parabola is \(\brac{-\frac{b}{2a}, f\brac{-\frac{b}{2a}}}\text{.}\) This is made clear using calculus.
Determining critical points, \(f'(x) = 2ax + b = 0\text{,}\) so \(x = -\frac{b}{2a}\text{.}\) Using the first derivative test, this is a maximum or minimum, depending on the sign of \(a\text{.}\)
Determining concavity, \(f''(x) = 2a\) which is constant. If \(a > 0\text{,}\) then \(f''(x) > 0\) for all \(x\text{,}\) and \(f\) is concave up. If \(a \lt 0\text{,}\) then \(f'' \lt 0\text{,}\) and \(f\) is concave down.
Subsection 4.6.4 Cubic Polynomials
Note that a cubic function always has exactly one point of inflection, because the second derivative is a linear function, which has one root.
Note that a polynomial function of degree \(n\) can have at most \(n-1\) relative extrema, and at most \(n-2\) inflection points.
If a polynomial has a double root, then it has a horizontal tangent there (like the function \(f(x) = x^3\)).
Subsection 4.6.5 Sketching the Graph of a Rational Function
The procedure for a rational function is somewhat similar to that of a polynomial function, except with rational functions we have a few extra characteristics.
- Rational functions can have values of \(x\) where \(f'(x)\) does not exist (critical numbers), or where \(f''(x)\) does not exist.
- Rational functions have can have horizontal and vertical asymptotes, and may have holes.
