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5.6 – Graphing Polynomial Functions

Key Terms

  • Concavity – A description of the direction of a graph’s curve.
  • Inflection Point – A point at which a graph’s concavity changes.



  • End Behavior of a Graph
    • How a graph starts and ends
    • See Algebra 2B, Section 5.2


  • Large and Small Numbers
    • Numbers get “smaller” as you move to the left on the number line.
    • Numbers get “larger” as you move to the right on the number line.
    • Numbers close to zero are not necessarily small.
      • Ex. Zero is smaller than 4, and -10 is smaller than zero.
      • Ex. Zero is larger than -10, and 4 is larger than zero.
      • Ex. 430 > -435,351,010
      • Ex. -500 < 0.00000000000000000000001


  • Multiplicity – When one factor appears more than once
    • Ex. f(x)=x^3-3x-2
    • f(x)=(x-2)(x+1)(x+1)
    • f(x)=(x-2)(x+1)^2
    • The y-intercept is: (0,-2)
    • The x-intercepts are: (2, 0) and (-1, 0)
      • (-1, 0) is a double root
    • In the following examples, notice that the graph will touch the x-axis at an extreme or at a curve

Alg2B 5.6 Multiplicity


  • Maximums and Minimums
    • Relative maxima are sections of the graph that look like hills, but are not the highest point(s) on the graph.
    • Relative minima are sections of the graph that look like valleys, but are not the lowest point(s) on the graph.
    • Absolute maximum – the highest point on the graph (the tallest hill)
    • Absolute minimum – the lowest point on the graph (the lowest valley)



  • End Behaviors
    • If the degree of a function is even, the end behaviors are the same (both up or both down).
    • If the degree of a function is odd, the end behaviors are different (one is up and one is down).

Alg2B 5.6 End Behavior

  • Graph Patterns
    • Positive Leading Coefficient & Even Power: Both Ends UP
    • Negative Leading Coefficient & Even Power: Both Ends DOWN
    • Positive Leading Coefficient & Odd Power: Left DOWN, Right UP
    • Negative Leading Coefficient & Odd Power: Left UP, Right DOWN


  • Notation
    • As the x-values get very small, the function, f(x), gets very small
      • x \rightarrow -\infty
      • f(x) \rightarrow -\infty
    • As the x-values get very large, the function, f(x), gets very large
      • x \rightarrow \infty
      • f(x) \rightarrow \infty


  • Y-Intercepts
    • When x = 0, you can find the y-intercept
    • Notation: (0, c)
    • Where a graph crosses the y-axis


  • Video: Finding x and y Intercepts and Graphing the Function


  • More About Multiplicity
    • When the multiplicity is odd, the graph crosses the x-axis.
      • (x-4)^3
      • (x+1)(x-4)^3
    • When the multiplicity is even, the graph touches the x-axis.
      • (x-3)^2
      • (x+2)(x-3)^2
    • Remember, exponents (powers) represent x-intercepts and changes in direction.
      • Multiplicity is impossible to see in a graph, as it doesn’t change direction, and it only crosses or touches the x-axis one time (even if there are many of the same root).
      • Try the practice on Pg 12 before taking the quiz, seriously!


  • Extreme Values
    • Extreme values are valleys and hills seen in graphs.
    • If a polynomial function has degree n, the graph will have at most n − 1 relative maximum or relative minimum values.
      • Ex. if a polynomial function has a degree of 4, its graph has at most 4 − 1, which is equal to 3, extreme values.  It may have 1, 2, or 3 extreme values.  It cannot have more than 3, but it can have less.
    • Graphs that continue up or down for infinity do NOT have any extreme values
      • They may relative maximums and minimums, but not absolute (or extreme).
    • Extremes are denoted by the y-coordinate.
      • Ex. If a relative maximum has its highest point at (3, 4), the relative maximum is 4 while the x = 3.
      • Ex. if a relative minimum has its lowest point at (3, -4), the relative minimum is -4 while x = 3.


  • Inflection Point


  • Graphing Completely



  • Ex 1. Which of the following correctly describes the end behavior of the polynomial function:
    • f(x)=-x^3+x^2-4x+2
    • Answer: Left end up, right end down



  • Ex 2. Which of the following notations correctly describe the end behavior of the polynomial graphed below?
    • Domain: notice that the graph is a parabola, with x-values going forever in the left (negative) and right (positive) directions.
    • It doesn’t matter that it is flipped over the x-axis.Alg2B 5.6 Ex 02
    • Answer:
    • x \rightarrow -\infty, f(x) \rightarrow -\infty
    • x \rightarrow \infty, f(x) \rightarrow \infty



  • Ex 3. What are the x-intercepts of the function:
    • f(x)=x^2-81?
      • The roots ARE the x-intercepts.
      • Since this quadratic function can be factored into: (x-9)(x+9), you can solve for the roots: 9 and -9.



  • Ex 4. Which of the following is an x-intercept of the function:
    • f(x)=x^3+3x^2-10x-24?
    • Answer Possibilities:  a. -4, b. -3, c. 2, d. 4
      • Test each one using substitution.
      • Correct answer: -4
        • f(-4)=(-4)^3+3(-4)^2-10(-4)-24
        • f(-4)=-64+3(16)-10(-4)-24
        • f(-4)=-64+48+40-24
        • f(-4)=0
          • Which means -4 is a root



  • Ex 5. Which polynomial function is graphed below?
    • Notice that this is an s-curve, so it is probably a cubic.Alg2B 5.6 Ex 05
    • Answer:
      • f(x)=(x+3)^2(x-1)
      • If you expand the factors, you get:
        • f(x)=(x+3)(x+3)(x-1)
        • f(x)=(x^2+6x+9)(x-1)
        • f(x)=(x^3+6x^2+9x-x^2-6x-9)
        • f(x)=x^3+5x^2+3x-9
          • This is a cubic graph with a positive leading coefficient, so it starts low and ends high.



  • Ex 6. What is the maximum number of possible extreme values for the function:
    • f(x)=x^3+4x^2-3x-18?
      • There is always 1 less extreme value than there are roots (see the largest exponent).
      • Answer: 2



  • Ex 7. Which point is the best approximation of the relative maximum of the polynomial function graphed below?
    • Look for the point where there is a hill. What is the coordinate (x, y) of that hill’s top?
    • Also notice that the x-values are marked by increments of 2, and the y-values are marked by increments of 5.Alg2B 5.6 Ex 07
    • Answer: (–3.6, 17)



  • Ex 8. How many roots does the polynomial function, y=(x-8)(x+3)^2 have?
    • Since 2 of the x-values are the same, they would have the same exact root (a double root).
    • So, the roots are: 8 and -3.
    • Answer: There are 2 unique roots.


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