polynomial functions: graphs, applications, and models

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Objectives Students will learn how to:. Graph functions ( x ) = ax n Graph General Polynomial Functions Find Turning Points and End Behavior Intermediate Value Theorems Approximate Real Zeros Using Graphing Calculator - PowerPoint PPT Presentation

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  • Polynomial Functions: Graphs, Applications, and ModelsGraph functions (x) = axn

    Graph General Polynomial Functions

    Find Turning Points and End Behavior

    Intermediate Value Theorems

    Approximate Real Zeros Using GraphingCalculator

    Use Graphing Calculator to Determine Polynomial Models and Curve FittingObjectives Students will learn how to:

  • Example 1GRAPHING FUNCTIONS OF THE FORM (x) = axnSolution Choose several values for x, and find the corresponding values of (x), or y. a.Graph the function.

    x(x) 2 8 1 1001128

  • Example 1GRAPHING FUNCTIONS OF THE FORM (x) = axnSolution The graphs of (x) = x3 and g(x) = x5 are both symmetric with respect to the origin. b.Graph the function.

    xg(x) 1.57.6 1 100111.57.6

  • Example 1GRAPHING FUNCTIONS OF THE FORM (x) = axnSolution The graphs of (x) = x3 and g(x) = x5 are both symmetric with respect to the origin. c.Graph the function.

    xg(x) 1.511.4 1100111.511.4

    x(x) 216 110011216

  • Graphs of General Polynomial FunctionsAs with quadratic functions, the absolute value of a in (x) = axn determines the width of the graph. When a> 1, the graph is stretched vertically, making it narrower, while when 0 < a < 1, the graph is shrunk or compressed vertically, so the graph is broader. The graph of (x) = axn is reflected across the x-axis compared to the graph of (x) = axn.

  • Graphs of General Polynomial FunctionsCompared with the graph of the graph of is translated (shifted) k units up if k > 0 andkunits down if k < 0. Also, when compared with the graph of the graph of (x) = a(x h)n is translated h units to the right if h > 0 and hunits to the left if h < 0.The graph of shows a combination of these translations. The effects here are the same as those we saw earlier with quadratic functions.

  • Example 2EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONSSolution The graph will be the same as that of (x) = x5, but translated 2 units down. a.Graph the function.

  • Example 2EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONSSolution In (x) = (x + 1)6, function has a graph like that of (x) = x6, but since x + 1 = x ( 1), it is translated 1 unit to the left. b.Graph the function.

  • Example 2EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONSSolution The negative sign in 2 causes the graph of the function to be reflected across the x-axis when compared with the graph of (x) = x3. Because 2> 1, the graph is stretched vertically as compared to the graph of (x) = x3. It is also translated 1 unit to the right and 3 units up.c.Graph the function.

  • Unless otherwise restricted, the domain of a polynomial function is the set of all real numbers. Polynomial functions are smooth, continuous curves on the interval ( , ). The range of a polynomial function of odd degree is also the set of all real numbers. Typical graphs of polynomial functions of odd degree are shown in next slide. These graphs suggest that for every polynomial function of odd degree there is at least one real value of x that makes (x) = 0 . The real zeros are the x-intercepts of the graph.

  • Odd Degree

  • A polynomial function of even degree has a range of the form ( , k] or [k, ) for some real number k. Here are two typical graphs of polynomial functions of even degree.Even Degree

  • Recall that a zero c of a polynomial function has as multiplicity the exponent of the factor x c. Determining the multiplicity of a zero aids in sketching the graph near that zero. If the zero has multiplicity one, the graph crosses the x-axis at the corresponding x-intercept as seen here.

  • If the zero has even multiplicity, the graph is tangent to the x-axis at the corresponding x-intercept (that is, it touches but does not cross the x-axis there).

  • If the zero has odd multiplicity greater than one, the graph crosses the x-axis and is tangent to the x-axis at the corresponding x-intercept. This causes a change in concavity, or shape, at the x-intercept and the graph wiggles there.

  • Turning Points and End BehaviorThe previous graphs show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.

  • A polynomial function of degree n has at most n 1 turning points, with at least one turning point between each pair of successive zeros.

  • End BehaviorThe end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree. A polynomial of the formhas the same end behavior as .

  • End BehaviorFor instance,has the same end behavior as . It is large and positive for large positive values of x and large and negative for negative values of x with large absolute value.

  • End BehaviorThe arrows at the ends of the graph look like those of the graph shown here; the right arrow points up and the left arrow points down.The graph shows that as x takes on larger and larger positive values, y does also. This is symbolized as read as x approaches infinity, y approaches infinity.

  • End BehaviorFor the same graph, as x takes on negative values of larger and larger absolute value, y does also:

    as

  • End BehaviorFor this graph, we have

    as

    and as

  • Suppose that axn is the dominating term of a polynomial function of odd degree.If a > 0, then as and as Therefore, the end behavior of the graph is of the type that looks like the figure shown here.

    We symbolize it as .

  • Suppose that axn is the dominating term of a polynomial function of odd degree.2. If a < 0, then as and as

    Therefore, the end behavior of the graph looks like the graph shown here.

    We symbolize it as .

  • Suppose that axn is the dominating term of a polynomial function of even degree.If a > 0, then as Therefore, the end behavior of the graph looks like the graph shown here.

    We symbolize it as .

  • Suppose that is the dominating term of a polynomial function of even degree.2. If a < 0, then as Therefore, the end behavior of the graph looks like the graph shown here.

    We symbolize it as .

  • Example 3DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIALMatch each function with its graph.Solution Because is of even degree with positive leading coefficient, its graph is C. A.B.C.D.

  • Example 3DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIALMatch each function with its graph.Solution Because g is of even degree with negative leading coefficient, its graph is A. A.B.C.D.

  • Example 3DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIALMatch each function with its graph.Solution Because function h has odd degree and the dominating term is positive, its graph is in B. A.B.C.D.

  • Example 3DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIALMatch each function with its graph.Solution Because function k has odd degree and a negative dominating term, its graph is in D. A.B.C.D.

  • Graphing TechniquesWe have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function will show the following characteristics:1. all x-intercepts (zeros)2. the y-intercept3. the sign of (x) within the intervals formed by the x-intercepts, and all turning points4. enough of the domain to show the end behavior.

    In Example 4, we sketch the graph of a polynomial function by hand. While there are several ways to approach this, here are some guidelines.

  • Let be a polynomial function of degree n. To sketch its graph, follow these steps.Step 1 Find the real zeros of . Plot them as x-intercepts.Step 2 Find (0). Plot this as the y-intercept.

  • Step 3 Use test points within the intervals formed by the x-intercepts to determine the sign of (x) in the interval. This will determine whether the graph is above or below the x-axis in that interval.

  • Use end behavior, whether the graph crosses, bounces on, or wiggles through the x-axis at the x-intercepts, and selected points as necessary to complete the graph.

  • Synthetic DivisionSolve Equations of Higher Order Than Quadratics by Factoring

    Use Synthetic Division to Divide Polynomials

    Evaluate Polynomial Functions Using the Remainder Theorem (synthetic substitution)

    Test Potential Zeros (Roots, Solutions) Objectives Students will learn how to:

  • Solving Higher Degree Equations by FactoringThe equation x3 + 8 = 0 that follows is called a cubic equation because of the degree 3 term. Some higher-degree equations can be solved using factoring and/or the quadratic formula.

  • SOLVING A CUBIC EQUATIONSolve Solution Factor as a sum of cubes.orZero-factor propertyorQuadratic formula; a = 1, b = 2, c = 4

  • SOLVING A CUBIC EQUATIONSolve Solution Simplify.Simplify the radical.Factor out 2 in the numerator.

  • SOLVING A CUBIC EQUATIONSolve Solution Lowest terms

  • SolveFactor out common factor

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