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5.3 Graphs of polynomial functions  (Page 5/13)

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Sketching the graph of a polynomial function

Sketch a graph of   f ( x ) = −2 ( x + 3 ) 2 ( x − 5 ) .

This graph has two x -intercepts. At   x = −3 ,   the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x -intercept. At   x = 5 ,   the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y -intercept is found by evaluating   f ( 0 ) .

f ( 0 ) = −2 ( 0 + 3 ) 2 ( 0 − 5 ) = −2 ⋅ 9 ⋅ ( −5 ) = 90

The y -intercept is   ( 0 , 90 ) .

Additionally, we can see the leading term, if this polynomial were multiplied out, would be   − 2 x 3 ,   so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See [link] .

Showing the distribution for the leading term.

To sketch this, we consider that:

  • As   x → − ∞   the function   f ( x ) → ∞ ,   so we know the graph starts in the second quadrant and is decreasing toward the   x - axis.
  • Since   f ( − x ) = −2 ( − x + 3 ) 2 ( − x – 5 )   is not equal to   f ( x ) ,   the graph does not display symmetry.
  • At   ( − 3 , 0 ) ,   the graph bounces off of the x -axis, so the function must start increasing.

    At   ( 0 , 90 ) ,   the graph crosses the y -axis at the y -intercept. See [link] .

Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at   ( 5 , 0 ) .   See [link] .

Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).

As   x → ∞   the function   f ( x ) → −∞ ,   so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

Using technology, we can create the graph for the polynomial function, shown in [link] , and verify that the resulting graph looks like our sketch in [link] .

Graph of f(x)=-2(x+3)^2(x-5).
The complete graph of the polynomial function   f ( x ) = − 2 ( x + 3 ) 2 ( x − 5 )
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Sketch a graph of   f ( x ) = 1 4 x ( x − 1 ) 4 ( x + 3 ) 3 .

Graph of f(x)=(1/4)x(x-1)^4(x+3)^3.

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Using the intermediate value theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x -axis, we can confirm that there is a zero between them. Consider a polynomial function   f   whose graph is smooth and continuous. The Intermediate Value Theorem    states that for two numbers   a   and   b   in the domain of   f , if   a < b   and f ( a ) ≠ f ( b ) , then the function   f   takes on every value between   f ( a )   and   f ( b ) .   (While the theorem is intuitive, the proof is actually quite complicated and requires higher mathematics.) We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function   f   at   x = a   lies above the   x - axis and another point at   x = b   lies below the   x - axis, there must exist a third point between   x = a   and   x = b   where the graph crosses the   x - axis. Call this point   ( c ,   f ( c ) ) .   This means that we are assured there is a solution   c   where f ( c ) = 0.

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the   x - axis. [link] shows that there is a zero between   a   and   b .  

Graph of an odd-degree polynomial function that shows a point f(a) that’s negative, f(b) that’s positive, and f(c) that’s 0.
Using the Intermediate Value Theorem to show there exists a zero.

Intermediate value theorem

Let   f   be a polynomial function. The Intermediate Value Theorem    states that if   f ( a )   and   f ( b )   have opposite signs, then there exists at least one value   c   between   a   and   b   for which   f ( c ) = 0.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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