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2.2.2: Graphs of Polynomials Using Zeros

  • Page ID
    14193
  • Graphs of Polynomials Using Zeros

    How is finding and using the zeroes of a higher-degree polynomial related to the same process you have used in the past on quadratic functions?


    Graphing Polynomials Using Zeros

    The following procedure can be followed when graphing a polynomial function.

    • Use the leading-term test to determine the end behavior of the graph.
    • Find the x−intercept(s) of f(x) by setting f(x)=0 and then solving for x.
    • Find the y−intercept of f(x) by setting y=f(0) and finding y.
    • Use the x−intercept(s) to divide the x−axis into intervals and then choose test points to determine the sign of f(x) on each interval.
    • Plot the test points.
    • If necessary, find additional points to determine the general shape of the graph.

    The Leading-Term Test

    If anxn is the leading term of a polynomial. Then the behavior of the graph as x→∞ or x→−∞ can be known by one the four following behaviors:

    1. If an>0 and n even: f-d_7476d0bdd91ce2f26e3237195491315ecd378b8b61a9efac88e9708f+IMAGE_TINY+IMAGE_TINY.jpg 

    2. If an<0 and n even: f-d_d7d5e0905838171941c98b5d7400249f875c025e564f361b2a215907+IMAGE_TINY+IMAGE_TINY.jpg

    3. If an>0 and n odd: f-d_192341c14f2284951a398f20e62b4c75b8af0317d93d383b5514b53e+IMAGE_TINY+IMAGE_TINY.jpg

    4. If an<0 and n odd: f-d_953895f7aa2a92e3fa4275f1973c297b95e8105a17f0791b0497ddef+IMAGE_TINY+IMAGE_TINY.jpg


    Examples

    Example 1

    Earlier, you were asked to identify some similarities in graphing using zeroes between quadratic functions and higher-degree polynomials.

    Solution

    Despite the more complex nature of the graphs of higher-degree polynomials, the general process of graphing using zeroes is actually very similar. In both cases, your goal is to locate the points where the graph crosses the x or y axis. In both cases, this is done by setting the y value equal to zero and solving for x to find the x axis intercepts, and setting the x value equal to zero and solving for y to find the y axis intercepts.

    Example 2

    Find the roots (zeroes) of the polynomial:

    h(x)=x3+2x2−5x−6

    Solution

    Start by factoring:

    h(x)=x3+2x2−5x−6=(x+1)(x−2)(x+3)

    To find the zeros, set h(x)=0 and solve for x.

    (x+1)(x−2)(x+3)=0

    This gives

    x+1=0

    x−2=0

    x+3=0

    or

    x=-1

    x=2

    x=-3

    So we say that the solution set is {−3,−1,2}. They are the zeros of the function h(x). The zeros of h(x) are the x−intercepts of the graph y=h(x) below.

    f-d_0f138b93a77ecc309835d61786e5b9606aabcf46198d92c1b97d9e14+IMAGE_TINY+IMAGE_TINY.jpg

    Example 3

    Find the zeros of g(x)=−(x−2)(x−2)(x+1)(x+5)(x+5)(x+5).

    Solution

    The polynomial can be written as

    g(x)=−(x−2)2(x+1)(x+5)3

    To solve the equation, we simply set it equal to zero

    −(x−2)2(x+1)(x+5)3=0

    this gives

    x−2=0

    x+1=0

    x+5=0

    or

    x=2

    x=-1

    x=-5

    Notice the occurrence of the zeros in the function. The factor (x−2) occurred twice (because it was squared), the factor (x+1) occurred once and the factor (x+5) occurred three times. We say that the zero we obtain from the factor (x−2) has a multiplicity k=2 and the factor (x+5) has a multiplicity k=3.

    Example 4

    Graph the polynomial function f(x)=−3x4+2x3.

    Solution

    Since the leading term here is −3x4 then an=−3<0, and n=4 even. Thus the end behavior of the graph as x→∞ and x→−∞ is that of Box #2, item 2.

    We can find the zeros of the function by simply setting f(x)=0 and then solving for x.

    −3x4+2x3=0

    −x3(3x−2)=0

    This gives

    x=0 or x=\(\ 2\over 3\)

    So we have two x−intercepts, at x=0 and at x=\(\ 2\over 3\), with multiplicity k=3 for x=0 and multiplicity k=1 for x=\(\ 2\over 3\)  

    To find the y−intercept, we find f(0), which gives

    f(0)=0

    So the graph passes the y−axis at y=0.

    Since the x−intercepts are 0 and \(\ 2\over 3\), they divide the x−axis into three intervals: (−∞, 0), (0, \(\ 2\over 3\)), and (\(\ 2\over 3\), ∞). Now we are interested in determining at which intervals the function f(x) is negative and at which intervals it is positive. To do so, we construct a table and choose a test value for x from each interval and find the corresponding f(x) at that value.

    Interval Test Value x f(x) Sign of f(x) Location of points on the graph
    (−∞, 0) -1 -5 - below the x−axis
    (0, \(\ 2\over 3\)) \(\ 1\over 2\) \(\ 1\over 16\) + above the x−axis
    (\(\ 2\over 3\), ∞) 1 -1 - below the x−axis

    Those test points give us three additional points to plot: (−1, −5), (\(\ 1\over 2\), \(\ 1\over 16\)), and (1, -1). Now we are ready to plot our graph. We have a total of three intercept points, in addition to the three test points. We also know how the graph is behaving as x→−∞ and x→+∞. This information is usually enough to make a rough sketch of the graph. If we need additional points, we can simply select more points to complete the graph.

    f-d_590aaf0743484f5467d74aff20c997ba645d0ed74d625c8fdb56c434+IMAGE_TINY+IMAGE_TINY.jpg

    Example 5

    Find the zeros and sketch a graph of the polynomial

    f(x)=x4−x2−56

    Solution

    This is a factorable equation,

    f(x)=x4−x2−56

    =(x2−8)(x2+7)

    Setting f(x)=0,

    (x2−8)(x2+7)=0

    the first term gives

    x2−8=0

    x2=0

    x= ±\(\ \sqrt{8}\)

    = ±\(\ 2\sqrt{2}\)

    and the second term gives

    x2+7=0

    x2=−7

    x=±\(\ \sqrt{-7}\)

    =±\(\ i\sqrt{7}\)    

    So the solutions are ±\(\ 2\sqrt{2}\) and ±\(\ i\sqrt{7}\), a total of four zeros of f(x). Keep in mind that only the real zeros of a function correspond to the x−intercept of its graph. 

    f-d_27935087168664c693fc246cd15f9270d6505d9ce362a0808dffb2cd+IMAGE_TINY+IMAGE_TINY.jpg

    Example 6

    Graph g(x)=−(x−2)2(x+1)(x+5)3.

    Solution

    Use the zeros to create a table of intervals and see whether the function is above or below the x−axis in each interval:

    Interval Test value x g(x) Sign of g(x) Location of graph relative to x−axis
    (−∞, −5) -6 320 + Above
    x=−5 -5 0 NA  
    (-5, -1) -2 144 + Above
    x=−1 -1 0 NA  
    (-1, 2) 0 -100 - Below
    x=2 2 0 NA  
    (2, ∞) 3 -256 - Below

    Finally, use this information and the test points to sketch a graph of g(x).

    f-d_1b451b508e6bfb32fe8a13dc2dd5c525d184bebf9bc9a1f32f939921+IMAGE_TINY+IMAGE_TINY.jpg

    Review

    1. If c is a zero of f, then c is a/an _________________________ of the graph of f.
    2. If c is a zero of f, then (x - c) is a factor of ___________________?
    3. Find the zeros of the polynomial: P(x)=x3−5x2+6x

    Consider the function: f(x)=−3(x−3)4(5x−2)(2x−1)3(4−x)2.

    1. How many zeros (x-intercepts) are there?
    2. What is the leading term?

    Find the zeros and graph the polynomial. Be sure to label the x-intercepts, y-intercept (if possible) and have correct end behavior. You may use technology for questions 9-12.

    1. P(x)=−2(x+1)2(x−3)
    2. P(x)=x3+3x2−4x−12
    3. f(x)=−2x3+6x2+9x+6
    4. f(x)=−4x2−7x+3
    5. f(x)=2x5+4x3+8x2+6x
    6. f(x)=x4−3x2
    7. g(x)=x2−|x|
    8. Given: P(x)=(3x+2)(x−7)2(9x+2)3

    State:

    1. The leading term:
    2. The degree of the polynomial:
    3. The leading coefficient:

    Determine the equation of the polynomial based on the graph:

    1. f-d_a2c9f9878050e516e56a0eefb61d8313e702f6d50ab3d6df240aafad+IMAGE_TINY+IMAGE_TINY.png
    2. f-d_7e339675765d8aa3438a350c9e7e16eeadce092f4fd7b79435441ac0+IMAGE_TINY+IMAGE_TINY.png

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 2.4. 


    Vocabulary

    Term Definition
    Cubic Function A cubic function is a function containing an x3 term as the highest power of x.
    Intercept The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.
    interval An interval is a specific and limited part of a function.
    Leading-Term Test The leading-term test is a test to determine the end behavior of a polynomial function.
    Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
    Polynomial Graph A polynomial graph is the graph of a polynomial function. The term is most commonly used for polynomial functions with a degree of at least three.
    Quartic Function A quartic function is a function f(x) containing an x4 term as the highest power of ''x''.
    Roots The roots of a function are the values of x that make y equal to zero.
    Zeroes The zeroes of a function f(x) are the values of x that cause f(x) to be equal to zero.