polinomials functions
TRANSCRIPT
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3.1
Quadratic Functions and Models
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A quadratic function is a function of the form:
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a > 0Opens up
Vertex is lowest point
Axis of symmetry
Graphs of a quadratic function f ( x) = ax 2 + bx + c
a < 0Opens down
Vertex is highest point
Axis of symmetry
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Steps for Graphing a Quadratic Function
by Hand Determine the vertex. Determine the axis of symmetry.
Determine the y-intercept, f (0) . Determine how many x-intercepts the graph has. If there are no x-intercepts determine another
point from the y-intercept using the axis of
symmetry. Graph.
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Without graphing, locate the vertex and find the axis of symmetryof the following parabola. Does it open up or down?
Vertex:
Since -3 < 0 the parabola opens down.
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Finding the vertex by completing the square:
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10 0 10
15
15
(0,0)
(2,4)
y x2
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10 0 10
15
15
(0,0)
(2, -12)
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10 0 10
15
15
(2, 0)
(4, -12)
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10 0 10
15
15
(2, 13)
Vertex
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Determine whether the graph opens up or down.Find its vertex, axis of symmetry, y-intercept, x-intercept.
x-coordinate of vertex:
Axis of symmetry:
y-coordinate of vertex:
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There are two x-intercepts:
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5 0
10
10
Vertex: (-3, -13)
(-5.55, 0) (-0.45, 0)
(0, 5)
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3.3Polynomial Functions and
Models
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A polynomial function is a function of the form
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Polynomial. Degree 2.
Not a polynomial.
Not a polynomial.
Determine which of the following arepolynomials. For those that are, state the degree.
(a)
(b)
(c)
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If f is a polynomial function and r is a real number
for which f (r )=0, then r is called a (real) zero of f , or root of f . If r is a (real) zero of f , then
(a) r is an x-intercept of the graph of f .
(b) ( x - r ) is a factor of f .
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Use the above to conclude that x = -1 and x = 4 arethe real roots (zeroes) of f .
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1 is a zero of multiplicity 2.
-3 is a zero of multiplicity 1.-5 is a zero of multiplicity 5.
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.
If r is a Zero or Odd Multiplicity
If r is a Zero or Even Multiplicity
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Theorem
If f is a polynomial function of degreen, then f has at most n-1 turning points.
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TheoremFor large values of x, either positiveor negative, the graph of thepolynomial
resembles the graph of the power function.
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For the polynomial
(a) Find the x- and y-intercepts of the graph of f .
(b) Determine whether the graph crosses ortouches the x-axis at each x-intercept.
(c) Find the power function that the graph of f resembles for large values of x.
(d) Determine the maximum number of turningpoints on the graph of f .
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For the polynomial
(e) Use the x-intercepts and test numbers to findthe intervals on which the graph of f is above the x-axis and the intervals on which the graph is
below the x-axis.(f) Put all the information together, and connectthe points with a smooth, continuous curve toobtain the graph of f .
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(b) -4 is a zero of multiplicity 1. (crosses)-1 is a zero of multiplicity 2. (touches)5 is a zero of multiplicity 1. (crosses)
(d) At most 3 turning points.
(a) The x-intercepts are -4, -1, and 5.y-intercept:
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Test number: -5
f (-5) 160Graph of f : Above x-axis
Point on graph: (-5, 160)
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Test number: -2
f (-2) -14
Graph of f : Below x-axis
Point on graph: (-2, -14)
-4 < x
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Test number: 0 f (0) -20
Graph of f : Below x-axis
Point on graph: (0, -20)
-1 < x < 5
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Test number: 6
f (6) 490
Graph of f : Above x-axis
Point on graph: (6, 490)
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8 6 4 2 0 2 4 6 8
300
100
100
300
500(6, 490)
(5, 0)(0, -20)
(-1, 0)
(-2, -14)(-4, 0)
(-5, 160)
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3.7The Real Zeros of a Polynomial
Function
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Theorem: Division Algorithm for Polynomials
If f ( x) and g( x) denote polynomial functions and if g( x) isnot the zero polynomial, then there are uniquepolynomial functions q( x) and r ( x) such that
where r ( x) is either the zero polynomial or apolynomial of degree less than that of g( x).
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Remainder Theorem
Let f be a polynomial function. If f ( x) is
divided by x - c, then the remainder is f (c).
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Find the remainder if
x + 3 = x - (-3)
is divided by x + 3.
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Factor Theorem
1. If f (c)= 0, then x - c is a factor of f ( x).
2. If x - c is a factor of f ( x) , then f (c)=0 .
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Use the Factor Theorem to determine whether the
function has the factor
(a) x + 3
(b) x + 4
x +3 is not a factor of f ( x).
x + 4 is a factor of f ( x).(b) f (-4) = 0
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Theorem Number of Zeros
A polynomial function cannot have
more zeros than its degree.
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Theorem Rational Zeros Theorem
Let f be a polynomial function of degree 1 orhigher of the form
where each coefficient is an integer. If p/q in the
lowest terms, is a rational zero of f , then p must bea factor of a 0 and q must be a factor of a n.
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List the potential rational zeros of
p:
q:
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Find the real zeros of
Factor f over the reals.
There are at most five zeros.
Write factors of -12 and 1 to obtain the potentialrational zeros.
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Thus, -3 is a zero of f and x + 3 is a factor of f .
Thus, -2 is a zero of f and x + 2 is a factor of f .
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Thus f(x) factors as :
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Theorem Bounds on Zeros
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Let f denote a continuous function. If a
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Use the Intermediate Value Theorem to showthat the graph of function
has an x-intercept in the interval [-3, -2].
f (-3) = -11.2 < 0
f (-2) = 1.8 > 0