unit 2: polynomial functions graphs of polynomial functions 2.2 jmerrill 2005 revised 2008
TRANSCRIPT
Learning Goal• To find zeros and use transformations to sketch
graphs of polynomial functions
• To use the Leading Coefficient Test to determine end behavior
Significant features• The graphs of polynomial functions are
continuous (no breaks—you draw the entire graph without lifting your pencil). This is opposed to discontinuous functions (remember piecewise functions?).
• This data is continuous as opposed to discrete.
Examples of Polynomials
Degree Name Example
0 Constant 5
1 Linear 3x+2
2 Quadratic X2 – 4
3 Cubic X3 + 3x + 1
4 Quartic -3x4 + 4
5 Quintic X5 + 5x4 - 7
Significant features• The graph of a polynomial function has only
smooth turns. A function of degree n has at most n – 1 turns. − A 2nd degree polynomial has 1 turn− A 3rd degree polynomial has 2 turns− A 5th degree polynomial has…
Cubic Parent Function
Draw the parent functions on the graphs.
f(x) = x3
X Y
-3 -27
-2 -8
-1 -1
0 0
1 1
2 8
3 27
Domain ,
Range - ,
X Y
-3 81
-2 16
-1 1
0 0
1 1
2 16
3 81
Quartic Parent Function
Draw the parent functions on the graphs.
f(x) = x4
Domain ,
Range 0,
Graph and TranslateStart with the graph of y = x3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right.
X Y
-3 -27
-2 -8
-1 -1
0 0
1 1
2 8
3 27
X Y
0 -54
1 -16
2 -2
3 0
4 2
5 16
6 54
3Equation 2 3y x
Domain ,
Range ,
X Y
-3 81
-2 16
-1 1
0 0
1 1
2 16
3 81
Graph and Translate
Start with the graph of y = x4. Reflect it across the x-axis. Translate it 2 units down.
X Y
-3 -83
-2 -18
-1 -3
0 -2
1 -3
2 -18
3 -83
4Equation 2y x
Domain ,
Range , 2 X Y
-3 -81
-2 -16
-1 -1
0 0
1 -1
2 -16
3 -81
Max/Min• A parabola has a
maximum or a minimum
• Any other polynomial function has a local max or a local min. (extrema)
Local max
Local min
max
min
Look at the root where the graph of f(x) crossed the x-axis. What was the power of the factor?
A. 3
B. 2
C. 1
Look at each root where the graph of a function“wiggled at” the x-axis. Were the powers even or odd?
A. Even
B. Odd
Look at each root where the graph of a function was tangent to the x-axis. What was the power of the factor?
A. 4
B. 3
C. 2
D. 1
Describe the end behavior of a function if a > 0 and n is even.A. Rise left, rise right
B. Fall left, fall right
C. Rise left, fall right
D. Fall left, rise right
Describe the end behavior of a function if a < 0 and n is even.A. Rise left, rise right
B. Fall left, fall right
C. Rise left, fall right
D. Fall left, rise right
Describe the end behavior of a function if a > 0 and n is odd.A. Rise left, rise right
B. Fall left, fall right
C. Rise left, fall right
D. Fall left, rise right
Describe the end behavior of a function if a < 0 and n is odd.A. Rise left, rise right
B. Fall left, fall right
C. Rise left, fall right
D. Fall left, rise right
Leading Coefficient Test• As x moves without bound to the left or right,
the graph of a polynomial function eventually rises or falls like this:
• In an odd degree polynomial:− If the leading coefficient is positive, the graph
falls to the left and rises on the right− If the leading coefficient is negative, the graph
rises to the left and falls on the right
• In an even degree polynomial:− If the leading coefficient is positive, the graph
rises on the left and right− If the leading coefficient is negative, the graph
falls to the left and right
End Behavior• If the leading coefficient of a polynomial
function is positive, the graph rises to the right.
y = x2 y = x3 + … y = x5 + …
Finding Zeros of a Function• If f is a polynomial function and a is a real
number, the following statements are equivalent:
• x = a is a zero of the function
• x = a is a solution of the polynomial equation f(x)=0
• (x - a) is a factor of f(x)
• (a, 0) is an x-intercept of f
Example• Find all zeros of f(x) = x3 – x2 – 2x
• Set function = 0 0 = x3 – x2 – 2x
• Factor 0 = x(x2 – x – 2)
• Factor completely 0 = x(x – 2)(x + 1)
• Set each factor = 0, solve 0 = x
0 = x – 2; so x = 2
0 = x +1; so x = -1
Multiplicity (repeated zeros)• How many roots? • How many roots?
3 roots; x = 1, 3, 3.4 roots; x = 1, 3, 3, 4.
3 is a double root. It is tangent to the x-axis
3 is a double root. It is tangent to the x-axis
Roots of Polynomials• How many roots? • How many roots?
5 roots: x = 0, 0, 1, 3, 3. 0 and 3 are double roots
3 roots; x = 2, 2, 2
Double roots
Double roots
(tangent)
Triple root – lies flat
then crosses
axis (wiggles)
Given Roots, Find a Polynomial Function• There are many correct solutions. Our solutions
will be based only on the factors of the given roots:
• Ex: Find a polynomial function with roots 2, 3, 3
• Turn roots into factors: f(x) = (x – 2)(x – 3)(x – 3)
• Multiply factors: f(x) = (x – 2)(x2 – 6x + 9)
• Finish multiplying: f(x) = x3 – 8x2 + 21x -18
Sketch graph
f(x) = (x - 4)(x - 1)(x + 2)
Step 1: Find zeros. zeros: 2,1, and 4Step 2: Mark the zeros on a number line.
Step 3: Determine end behavior
Step 4: Sketch the graph
Fall left, rise right
Roots: -3, 2 and 6Factors: (x+3), (x-2) and (x-6)
Factored Form:
f(x) = (x+3)(x-2)(x-6)
Write an equation.
Polynomial Form: f(x) = (x+3)(x2 – 8x + 12)
= x3 – 5x2 – 12x + 36
Write equation.
Zeros: -2, -1, 3 and 5
Factors: (x+2), (x+1), (x-3) and (x-5)
Factored Form:
f(x) = (x + 2)(x + 1)(x – 3)(x – 5)
Polynomial Form: