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5.2 – Polynomial Functions

Key Terms

  • Cubic Function – A polynomial function of degree 3.
    • The general form of a cubic function is F(x)=ax^3+bx^2+cd+d, where a, b, c, and d are real numbers and a is not equal to zero.
  • End Behavior – The way in which a graph of a function appears on the left and right sides of the Cartesian coordinate plane.
    • It is the overall appearance of the graph.
  • Estimate – To approximate or make a rough calculation.
  • Extreme Value – A value of the function y = f(x) that is the greatest or least value of y.
    • The greatest value is called the maximum, and the least value is called the minimum.
  • Polynomial Function – A function that can be defined by evaluating an algebraic expression with one or more terms.
    • None of the variables is in the denominator of a fraction, and any exponents are whole numbers.
  • Roots – Values for which a function equals zero.
    • The roots are also called zeros of the function.
    • Any x-value at which the graph of a function crosses the x-axis is a root of the function.

 


Review

  • Volume = length •  width • height

 

  • Graphs
    • Read from left to right
    • To graph a polynomial equation, plot some points and then connect them with a smooth curve.
    • When you multiply all the coefficients on the right side of an equation by -1, the graph gets flipped over the x-axis.

 


Notes

  • Evaluating Functions

Alg2B 5.2 - Evaluating Functions

 

  • Polynomial Graph Shapes
    • Quadratics (highest degree: 2) and Quartics (highest degree: 4): parabola shapes
      • Quadratics have a hill (when negative: flipped) or valley (when positive) in the graph
        • Ex. y=x^2Alg2B 5.2 Quadratic
      • Quartics also have a hill (when negative: flipped) or valley (when positive) in the graph, but they are wider with sharper curves
        • Ex. y=x^4Alg2B 5.2 Quartic
        • Quartics can also have multiple hills and valleys
          • Ex. y=x^4-2x^2Alg2B 5.2 Quartic 2
    • Cubics (highest degree: 3) and Quintics (highest degree: 5): S-shapes
      • Cubics are simple s-curves without hills or valleys
        • Ex. y=x^3Alg2B 5.2 Cubic
      • Quintics are also s-curves, but are wider with sharper curves
        • Ex. y=x^5Alg2B 5.2 Quintic
      • Both Cubics and Quintics can have multiple hills and valleys
        • Ex. y=x^3-2x^2Alg2B 5.2 Cubic 2

 

  • Polynomial Graph Properties
    • Polynomials that are “not in their simplest form” have parent functions have been shifted, stretched, compressed, or flipped.
    • The place where a polynomial’s graph crosses the y-axis (its y-intercept) is the equation’s constant.
    • All polynomials have smooth, continuous graphs that do not cross over themselves.
      • The plotted points must be connected by only one curving line without breaks.
      • There is only one way to connect the graph’s plotted points.

Alg2B 5.2 - Change Direction

 

  • Degree – the highest exponent value of a polynomial
    • Even degree: start high, end high (up)
    • Odd degree: start low, end high (up)
    • Positive leading coefficient: end high (up)
    • Negative leading coefficient: end low (down)

Alg2B 5.2 - Graph End Behavior

Alg2B 5.2 - Degree Graphs

 

  • The bumps (hills and valleys) in the graph are based on having additional x-variables in the polynomial
    • Ex. y=x^2 is a simple parabola with a valley, but some look like this: y=x^4+2x^3, and have a hill and two valleys (see Fourth-Degree Function below).
    • Ex. y=x^3 is a simple s-curve, but some look like this: y=x^3-2x$, and have a hill and a valley (see Third-Degree Function below).
    • Ex. Some polynomials have many hills and valleys: y=x^5-5x^4+5x^3+5x^2-5.5x-1 (see Fifth-Degree Function below)

 

  • Extreme Values
    • A value for which the polynomial is bigger or smaller than any nearby values.
    • A maximum looks like the top of a hill
    • A minimum looks like the bottom of a valley
    • An absolute maximum is the highest point on the graph or in a given domain
    • An absolute minimum is the lowest point on the graph or in a given domain
    • A relative maximum is a hill in the graph, but it’s not the highest hill
    • A relative minimum is a valley in the graph, but it’s not the lowest valley

Alg2B 5.2 - Max Min

  • Maximum and Minimum Values
    • The maximum of a polynomial is its greatest value and the highest point on its graph.
    • You can estimate the extreme value or values by plotting points.
      • First, plot enough points so you can see the shape of the graph.
      • Then look for the area where you’d expect to find the maximum value and try to figure out what range of x-values the points in that area would have.
      • The x-value that gives the greatest y-value is the best estimate for the graph’s maximum.

 


  • Roots of a Polynomial
    • Where the graph crosses the x-axis
    • The x-values for which y = 0
    • Also called zeros of the function
    • For higher-order polynomials, it is possible to have as many roots as the degree of the function.

 

  • Graphing to Estimate Roots
    • Step 1: Use the scale and graph to identify a range for each root.
    • Step 2: Try different x-values within that range.
    • Step 3: When you find the x-value that gives y = 0, you have found the root.

 


Examples

  • Ex 1. Evaluate the function below for x = 4.
    • F(x)=x^3+2x^2+1
    • Substitute 4 for all x values: F(4)=4^3+2(4)^2+1
    • Simplify: F(4)=64+32+1
    • Answer: F(4)=97

 

  • Ex 2. At what values of x, does f(x) = 0?
    • This is where all y-values equal 0 (the roots)

Alg2B 5.2 - RootsAnswers: -1, 2, and 4

 


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