Polynomial Functions and Graphing Guide

Polynomial Functions and Graph Sketching

Key Definitions

• X-intercepts (Intercept Group)

  • These are the points where the graph crosses or touches the x-axis.
  • To find them, set each factor of the polynomial function equal to zero and solve for $x$.
  • Sign Convention: When pulling an x-intercept out of a factor, always change its sign. For example, if a factor is $(x - 5)$, the x-intercept is $x = +5$.

• Vertex Points

  • Also known as turning points, humps, or bounces.
  • The number of vertex points a polynomial can have is at most one less than its degree (the highest exponent).
    • Linear Function ($x^1$): Has a degree of $1$, so it has at most $(1-1) = 0$ turning points.
    • Parabola ($x^2$): Has a degree of $2$, so it has at most $(2-1) = 1$ turning point.
    • Cubic Function ($x^3$): Has a degree of $3$, so it has at most $(3-1) = 2$ turning points.

• Maximum and Minimum Values

  • Absolute Maximum/Minimum: Refers to the highest or lowest $y$-value the graph reaches across its entire domain.
    • Odd-Powered Polynomials (e.g., cubic, quintic): The graph will extend infinitely in opposite directions (one end up, one end down). Therefore, the absolute minimum is $-\infty$ and the absolute maximum is $+\infty$.
    • Even-Powered Polynomials (e.g., parabola, quartic): Both ends of the graph will either go up or both will go down. This means there will be a definite absolute minimum or maximum value.
  • Local Maximum/Minimum: Refers to the highest or lowest $y$-value within a specific, restricted domain (a particular section of the graph).

Curve Sketching Procedure

Curve sketching is a method to approximate the shape of a polynomial graph, as precise plotting often requires computational tools.

1. Factor the Function: The function must be in its factored form (e.g., $(x-a)(x-b)(x-c)$) to easily identify x-intercepts and their multiplicity. If not provided in factored form, you will need to factor it first.

2. Find X-Intercepts: Set each individual factor equal to zero and solve for $x$. Remember the sign change rule (e.g., from $(x-5)$ to $x=+5$).

3. Determine Behavior at X-intercepts: The behavior of the graph at each x-intercept depends on the power (multiplicity) of its corresponding factor:

   • Odd Powered Factor (Multiplicity of $1$, $3$, $5$, etc.): The graph will go through the x-intercept (crosses the x-axis).
   • Even Powered Factor (Multiplicity of $2$, $4$, $6$, etc.): The graph will bounce off the x-intercept (touches the x-axis and turns back in the direction it came from). For this course, we will primarily focus on factors with powers of $1$ (odd) and $2$ (even).

4. Determine End Behavior: This describes how the graph behaves as $x$ approaches positive or negative infinity (the left-most and right-most parts of the graph). It depends on two things: the degree of the polynomial and the sign of the leading coefficient.

   • If the Polynomial Degree is ODD (e.g., cubic, quintic): The graph starts and ends in opposite directions.
     • Positive Leading Coefficient (e.g., $y=x^3$): The graph starts low (as $x \to -\infty$, $y \to -\infty$) and ends high (as $x \to +\infty$, $y \to +\infty$). (Starts at the opposite sign of the leading coefficient's value.)
     • Negative Leading Coefficient (e.g., $y=-x^3$): The graph starts high (as $x \to -\infty$, $y \to +\infty$) and ends low (as $x \to +\infty$, $y \to -\infty$).
   • If the Polynomial Degree is EVEN (e.g., quadratic, quartic): The graph starts and ends in the same direction.
     • Positive Leading Coefficient (e.g., $y=x^2$, $y=x^4$): The graph starts high (as $x \to -\infty$, $y \to +\infty$) and ends high (as $x \to +\infty$, $y \to +\infty$). (Starts at the same sign as the leading coefficient's value.)
     • Negative Leading Coefficient (e.g., $y=-x^2$): The graph starts low (as $x \to -\infty$, $y \to -\infty$) and ends low (as $x \to +\infty$, $y \to -\infty$).

5. Sketch the Graph: Combine the information from steps 2-4. Start with the correct end behavior, move towards the x-intercepts, applying the