graphs reveal key insights about a function's behavior. By examining , end behavior, and turning points, we can visualize how polynomials behave across different x-values. These features are crucial for understanding function characteristics and solving real-world problems.
Graphing polynomials involves factoring, determining end behavior, and identifying critical points. By mastering these techniques, we can predict a polynomial's shape and properties. This knowledge is essential for analyzing complex systems and making informed decisions in various fields.
Polynomial Function Graphs
Key features of polynomial graphs
Zeros (roots or x-intercepts)
Function intersects x-axis at these points
Occurs when function value equals zero (f(x)=0)
End behavior
Function behavior as x approaches positive or negative infinity
Determined by leading term's degree and coefficient
Even degree: both ends approach positive or negative infinity based on leading coefficient sign (positive coefficient means both ends approach positive infinity)
Odd degree: ends approach opposite infinities based on leading coefficient sign (positive coefficient means left end approaches negative infinity, right end approaches positive infinity)
Turning points (local maxima and minima)
Function changes direction at these points (increasing to decreasing or vice versa)
Occur at critical points where derivative equals zero or is undefined (f′(x)=0 or f′(x) is undefined)
Factoring for polynomial zeros
Factor polynomial to find zeros
Set function equal to zero and factor equation (f(x)=0)
Zeros are x-values that make factored expressions equal zero
Common factoring techniques
Greatest common factor (GCF)
Grouping
Difference of squares (a2−b2=(a+b)(a−b))
Sum and difference of cubes (a3+b3=(a+b)(a2−ab+b2) and a3−b3=(a−b)(a2+ab+b2))
Quadratic formula for quadratic factors (x=2a−b±b2−4ac)
Synthetic division can be used to efficiently find zeros of polynomials
Degree vs graph characteristics
Degree: highest power of variable in polynomial
Number of turning points
At most one less than polynomial degree
4th-degree polynomial can have at most 3 turning points
Number of real zeros
At most equal to polynomial degree
3rd-degree polynomial can have at most 3 real zeros
of zeros
Number of times zero is repeated in factored polynomial
Affects graph shape near zero (see "Zero multiplicity in graphs")
Graphing with polynomial properties
Steps to graph polynomial
Find zeros by factoring or using rational zero theorem
Determine end behavior from leading term's degree and coefficient
Identify y-intercept by evaluating function at x = 0 (f(0))
Find turning points by solving derivative for zero or undefined values (f′(x)=0 or f′(x) is undefined)
Plot zeros, y-intercept, and turning points
Connect points considering end behavior and shape near zeros based on multiplicity
Zero multiplicity in graphs
Multiplicity: number of times zero is repeated in factored polynomial
Effect on graph near zero
Odd multiplicity: graph crosses x-axis at zero
(x−1)3 has zero at x = 1 with multiplicity 3, graph crosses x-axis at x = 1
Even multiplicity: graph touches x-axis at zero but does not cross
(x−2)2 has zero at x = 2 with multiplicity 2, graph touches x-axis at x = 2 without crossing
Intermediate Value Theorem application
(IVT)
If polynomial f(x) is on closed interval [a,b] and k is between f(a) and f(b), then there exists c in [a,b] such that f(c)=k
Application to polynomials
Polynomials are continuous everywhere
IVT determines existence of zeros or specific function values within interval
Behavior prediction from algebraic form
Leading term
Determines function's end behavior
Coefficient and degree provide direction and rate of growth or decay
Zeros and multiplicity
Indicate x-axis intersections and graph shape near those points
Polynomial degree
Provides upper limit for number of turning points and real zeros
Constant term
Represents function's y-intercept (f(0))
Advanced Polynomial Analysis
Fundamental Theorem of Algebra: Every non-constant polynomial with complex coefficients has at least one complex root
Complex roots always occur in conjugate pairs for polynomials with real coefficients
Descartes' Rule of Signs: Determines the possible number of positive and negative real roots
Vieta's formulas: Relate the coefficients of a polynomial to sums and products of its roots
Key Terms to Review (10)
Continuous: A function is continuous if there are no breaks, holes, or jumps in its graph. This means you can draw the graph without lifting your pencil.
Global maximum: A global maximum is the highest point over the entire domain of a function. It is where the function reaches its greatest value.
Global minimum: A global minimum of a function is the lowest point over its entire domain. It is where the function attains its smallest value.
Intermediate Value Theorem: The Intermediate Value Theorem states that for any continuous function $f$ on the interval $[a, b]$, if $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one value $c$ in the interval $(a, b)$ such that $f(c) = N$. It guarantees the existence of roots within an interval where the function changes sign.
Local maximum: A local maximum is a point on the graph of a function where the function value is greater than or equal to the values of the function at nearby points. It represents a peak in a specific region of the graph.
Local minimum: A local minimum of a function is a point where the function value is lower than at all nearby points. It represents the lowest value within a specific interval or neighborhood.
Multiplicity: Multiplicity refers to the number of times a particular root is repeated in a polynomial function. It affects the shape and behavior of the graph at that root.
Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants.
Turning point: A turning point of a polynomial function is where the graph changes direction from increasing to decreasing or vice versa. It is often associated with local maxima and minima.
Zeros: Zeros of a function are the values of the variable that make the function equal to zero. For polynomial functions, they are also known as roots or solutions.