Evaluating and Graphing Polynomial Functions
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EVALUATING POLYNOMIAL FUNCTIONS A polynomial function is a function of the form f (x) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n 0 and the exponents are all whole numb polynomial function is in standard form if its te itten in descending order of exponents from left to For this polynomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n 0 a n a n leading coefficient a 0 a 0 constant term n n degree descending order of exponents from left to n n – 1
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Transcript of Evaluating and Graphing Polynomial Functions
EVALUATING POLYNOMIAL FUNCTIONS
A polynomial function is a function of the form
f (x) = an x n + an – 1 x
n – 1 +· · ·+ a 1 x + a 0
Where an 0 and the exponents are all whole numbers.
A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.
For this polynomial function, an is the leading coefficient,
a 0 is the constant term, and n is the degree.
an 0
an
an leading coefficient
a 0
a0 constant term n
n
degree
descending order of exponents from left to right.
n n – 1
Degree Type Standard Form
EVALUATING POLYNOMIAL FUNCTIONS
You are already familiar with some types of polynomialfunctions. Here is a summary of common types ofpolynomial functions.
4 Quartic f (x) = a4 x 4 + a 3 x
3 + a 2 x 2 + a 1 x + a 0
0 Constant f (x) = a 0
3 Cubic f (x) = a 3 x 3 + a 2 x
2 + a 1 x + a 0
2 Quadratic f (x) = a 2 x 2 + a 1 x + a 0
1 Linear f (x) = a1x + a 0
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
f (x) = x 2
– 3x4 – 712
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.
Its standard form is f (x) = – 3x 4
+ x 2 – 7. 1
2
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
Identifying Polynomial Functions
The function is not a polynomial function because the
term 3
x does not have a variable base and an exponentthat is a whole number.
SOLUTION
f (x) = x 3 + 3
x
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
SOLUTION
f (x) = 6x 2 + 2 x
–1 + x
The function is not a polynomial function because the term2x
–1 has an exponent that is not a whole number.
Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 2, so it is a quadratic function.
The leading coefficient is .
Its standard form is f (x) = x2 – 0.5x – 2.
f (x) = – 0.5 x + x 2 – 2
f (x) = x 2 – 3 x
4 – 712
Identifying Polynomial Functions
f (x) = x 3 + 3x
f (x) = 6x2 + 2 x– 1 + x
Polynomial function?
f (x) = – 0.5x + x2 – 2
Using Synthetic Substitution
One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.
Use synthetic division to evaluate
f (x) = 2 x 4 + 8 x
2 + 5 x 7 when x = 3.
Polynomial in standard form
Using Synthetic Substitution
2 x 4 + 0 x
3 + (–8 x 2) + 5 x + (–7)
2 6
6
10
18
35
30 105
98
The value of f (3) is the last number you write,In the bottom right-hand corner.
The value of f (3) is the last number you write,In the bottom right-hand corner.
2 0 –8 5 –7 CoefficientsCoefficients
3
x-value
3 •
SOLUTION
Polynomial instandard form
GRAPHING POLYNOMIAL FUNCTIONS
The end behavior of a polynomial function’s graphis the behavior of the graph as x approaches infinity(+ ) or negative infinity (– ). The expressionx + is read as “x approaches positive infinity.”
GRAPHING POLYNOMIAL FUNCTIONS
END BEHAVIOR
GRAPHING POLYNOMIAL FUNCTIONS
END BEHAVIOR FOR POLYNOMIAL FUNCTIONSCONCEPT
SUMMARY
> 0 even f (x) + f (x) +
> 0 odd f (x) – f (x) +
< 0 even f (x) – f (x) –
< 0 odd f (x) + f (x) –
an n x – x +
x
f(x)–3
–7
–2
3
–1
3
0
–1
1
–3
2
3
3
23
Graphing Polynomial Functions
Graph f (x) = x 3 + x
2 – 4 x – 1.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
The degree is odd and the leading coefficient is positive, so f (x) – as x – and f (x) + as x + .
x
f (x)
–3
–21
–2
0
–1
–1
0
0
1
3
2
–16
3
–105The degree is even and the leading coefficient is negative, so f (x) – as x – and f (x) – as x + .
Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x
3 + 2x 2 + 4x.
SOLUTION
To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.
Assignment
