Polynomial graphs are smooth curves that can model motion, area, revenue, and many other changing quantities. Graphing them helps you see where a function is positive, negative, increasing, decreasing, or equal to zero. The degree and leading coefficient give the first big clues about the overall shape.
Zeros, multiplicities, and turning points then refine the graph into a useful picture.
Key Facts
- A polynomial has the form f(x) = anx^n + a(n-1)x^(n-1) + ... + a1x + a0, where n is a nonnegative integer.
- The degree is the highest power of x with a nonzero coefficient.
- The leading coefficient is the coefficient of the highest-degree term, and it controls end behavior together with the degree.
- A zero is an x-value where f(x) = 0, so the graph has an x-intercept at that value if the zero is real.
- If a zero has odd multiplicity, the graph crosses the x-axis there; if it has even multiplicity, the graph touches and turns around there.
- A degree n polynomial can have at most n real zeros and at most n - 1 turning points.
Vocabulary
- Polynomial function
- A function made from sums of terms ax^k, where a is a real coefficient and k is a nonnegative integer.
- Degree
- The largest exponent of x in a polynomial after like terms are combined.
- Leading coefficient
- The coefficient of the term with the highest power of x.
- Zero
- An input value that makes the polynomial equal to 0.
- Multiplicity
- The number of times a factor is repeated in the factored form of a polynomial.
Common Mistakes to Avoid
- Using only the degree to decide end behavior is wrong because the leading coefficient also matters.
- Assuming every zero crosses the x-axis is wrong because zeros with even multiplicity make the graph touch the axis and turn around.
- Counting all bends as guaranteed turning points is wrong because a degree n polynomial has at most n - 1 turning points, not exactly n - 1.
- Ignoring vertical scale when sketching is wrong because the graph may have steep growth or shallow turning points that change how features appear.
Practice Questions
- 1 For f(x) = (x + 2)(x - 1)^2(x - 3), list the zeros, state each multiplicity, and say whether the graph crosses or touches the x-axis at each zero.
- 2 Determine the degree, leading coefficient, end behavior, and maximum possible number of turning points for g(x) = -2x^5 + 4x^3 - x + 7.
- 3 Explain how the graph of h(x) = (x - 4)^2(x + 1) should behave near x = 4 and why that behavior is different from its behavior near x = -1.