Chapter 3.4
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Transcript of Chapter 3.4
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Chapter 3.4
Polynomial Functions
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y
x
Graphy = x3
y = x5
y = x4
y = x6
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As with quadratic functions, the value of a in f(x) = axn determines the width of the graph.
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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.
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y
x
Graphy = x2
y = 2x2
y = 1/2x3
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The graph of f(x) = -axn is reflected across the x-axis compared to the graph of f(x) = -axn
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y
x
Graphy = x2
y = -x2
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Compared with the graph of f(x) = axn
the graph of f(x) = axn + k is translated (shifted) k units up if k > 0.
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Also, when compared with the graph of f(x) = axn
the graph of f(x) = a(x-h)n is translated h units to the right if k > 0 and |h| units to the left if k < 0.
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The graph of f(x) = a(x-h)n + k shows a combination of these translations. The effects here are the same as those we saw earlier with quadratic functions.
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y
x
Graphy = x5 - 2
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y
x
Graphy = (x+1)6
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y
x
Graphy = -2(x-1)3 +3
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The domain of every polynomial function is the set of all real numbers; thus polynomial functions are continuous on the interval (-∞, ∞).
The range of a polynomial function of odd degree is also the set of all real numbers.
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Typical graphs of polynomial functions of odd degree are shown in Figure 22. These graphs suggest that for every polynomial function f of odd degree there is at least one real value of x that makes f(x) = 0.
The zeros are the x-intercepts of the graph.
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A polynomial function of even degree has range of the form (-∞, k] or [k, ∞) for some real number k. Figure 23 shows two typical graphs of polynomial functions of even degree.
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22 )2(1)(x y :tyMultiplici x
axis.- x thecrosses
graph theodd, isexponent theIf
axis.- x thetouches
graph theeven, isexponent theIf
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The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree.
A polynomial of the formf(x) = anxn + an-1xn + . . . + a0 has the same endbehavior as f(x) = anxn.
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For instancef(x) = 2x3 – 8x2 + 9 has the same end behavior as f(x) = 2x3. It is large and positive for large positive values of x and large and negative for large negative values of x. with large absolute value.
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The arrows at the ends of the graph look like those of the first graph in Figure 22; the right arrow points up and the left arrow points down.
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The first graph in Figure 22 shows that as x takes on larger and larger positive values, y does also. This is symbolized
,y , xas
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For the same graph, as x takes on negative values of larger and larger absolute value, y does also.
.y , xas
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For the middle graph in Figure 22 we have
y , xas
y , xas
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Graphing TechniquesA comprehensive graph of a polynomial funciton
will show the following characteristics.
1. all x-intercepts2. the y-intercept3. all turning points4. enough of the domain to show the end
behavior.
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We emphasize the important relationships among
the following concepts.
1. the x-intercepts of the graph of y = f(x)2. the zeros of the function f3. the solutions of the equation f(x) = 04. the factors of f(x)
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Caution:Be careful how you interpret the intermediate value theorem.
If f(a) and f(b) are not opposite in sign, it does not necessarily mean that there is no zero between a and b.
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For example, in Figure 30, f(a) and f(b) are both negative, but -3 and -1, which are between a and b, are zeros of f(x).
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Use synthetic division to show that f(x) = x3 -2x2 – x + 1 has a real zero between 2 and 3.
Use synthetic division to find f(2) and f(3)
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Use synthetic division to show that f(x) = x3 -2x2 – x + 1 has a real zero between 2 and 3.
Use synthetic division to find f(2) and f(3)
Since f(2) is negative and f(3) is positive, by the intermediate value theorem there must be a real zero between 2 and 3.
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Show that the real zeros of f(x) = 2x4 -5x3 + 3x + 1satisfy the following conditions.
(a) No real zero is greater than 3
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Show that the real zeros of f(x) = 2x4 -5x3 + 3x + 1satisfy the following conditions.
(b) No real zero is less than -1