AP Precalculus · Lesson 1 of 4
Polynomial and rational functions
Connect factors to zeros, excluded inputs, holes, and end behavior.
Zeros and factors
If (x − 4) is a factor of a polynomial, then x = 4 is a zero and the graph meets the x-axis there. A repeated factor can change whether the graph crosses or touches the axis. The end behavior of a polynomial depends on the degree and the sign of the leading coefficient.
Evaluating p(2) is different from solving p(x) = 2. A table can reveal a constant difference for a linear function, or a constant second difference for a quadratic. Compare equal steps in the input before you name the family.
Rational functions
A rational function is undefined where its original denominator is zero. If a factor cancels, the graph has a hole at that input. If a factor remains in the denominator, the graph has a vertical asymptote there, unless that input was already excluded for another reason you must still state.
For f(x) = (x² − 9)/(x − 3), the simplified rule x + 3 agrees with f everywhere except x = 3. The graph has a hole at the point (3, 6), not a vertical asymptote.
Worked example
Describe the graph of f(x) = (x − 1)(x + 2)/(x − 1).
- The original function is undefined at x = 1.
- For every other x, f(x) = x + 2.
- The graph is the line y = x + 2 with a hole at (1, 3).
Why this works. Cancellation removes the factor from the simplified rule, but it does not put the excluded input back into the domain.
Check your understanding
- List zeros from factors and excluded inputs from the original denominator.
- Decide whether a canceled factor creates a hole or a remaining factor creates a vertical asymptote.
- Separate evaluating a function from solving an equation.