6.2 Evaluating & Graphing Polynomials
unction:: a function of the form f ( x ) = ax 4 + bx 3 + cx 2 + dx + e where the
polynomial ffunction
unction
exponents are all whole numbers and the coefficients are all real numbers.
coefficient:: the number in front of the x with the highest power
leading coefficient
term:: the number that has no x with it
constant term
degree: the variable with the highest exponent
degree
Ex:
Ex f ( x ) = x 2 + 2x + 1
f ( x ) = 3x 5 − 5x 3 + 7
form:: rearranging the polynomial so that its terms are written in descending
standard form
order of exponents from left to right
Degree
Type
Standard Form
Example
0
1
2
Constant
Linear
Quadratic
3
Cubic
4
Quartic
f (x ) = c
f ( x ) = ax + b
f ( x ) = ax 2 + bx + c
f ( x ) = ax 3 + bx 2 + cx + d
f ( x ) = ax 4 + bx 3 + cx 2 + dx + e
f ( x ) = −14
f ( x ) = 5x − 7
f ( x ) = 2x 2 + x − 9
f ( x ) = x 3 − x 2 + 3x − 6
f ( x ) = x 4 + 3x 3 − x 2 + 2x − 1
Ex:
Ex: Decide whether the function is a polynomial function. If it is, write the function
in standard form and state the degree, type, and leading coefficient.
1.) f ( x ) =
1 2
x − 3x 4 − 7
2
3.) f ( x ) = −0.5x + π x 2 − 2
2.) f ( x ) = x 3 + 3x
4.) f ( x ) = 6x 2 + 2x −1 + x
Direct Substitution
One way to evaluate a polynomial function is to use direct substitution. Simply
substitute the given value of x into the function in place of each x.
Ex:
Ex f ( x ) = 2x 4 − 8x 2 + 5x − 7 ; x = 3
Ex: f ( x ) = −3x 3 + x 2 − 12x − 5 ; x = 2
Synthetic Substitution
Substitution
1.) Write the coefficients of f(x) in order of descending exponents. Write the value at
which f(x) is being evaluated to the left.
2.) Bring down the leading coefficient. Multiply the leading coefficient by the xvalue. Write the product under the second coefficient. Add.
3.) Multiply the previous sum by the x-value. Write the product under the third
coefficient. Add. Repeat for all remaining coefficients. The final sum is the value
of f(x) at the given x-value.
Ex:
Ex f ( x ) = 2x 4 − 8x 2 + 5x − 7 ; x = 3
Ex: f ( x ) = −3x 3 + x 2 − 12x − 5 ; x = 2
Graphing Polynomial Functions
Ex:
Ex
Ex:
Ex
Ex:
Ex
f ( x ) = x 3 + 2x 2 − x + 3
x
f(x)
x
f(x)
x
f(x)
f ( x ) = −x 4 − 2x 3 + 2x 2 + 4x
f (x ) = x 5 − x 2 − 1