Download 2.1 Quadratic Functions and Models

2.1 Quadratic Functions and Models
What are we going to learn?
-analyzing graphs of quadratic functions
-writing quadratic functions in standard form
-using quadratic functions to model situations
Why are we learning it?
Quadratic functions can be used to model real world
situations.
Definition: polynomial function-
Definition: quadratic function-
Exploration: Transformations of parabolas
1. Graph y = ax2 for a = -2, -1, -0.5, 0.5, 1, and 2. How does changing the value of a affect the graph?
2. Graph y = (x – h)2 for h = -4, -2, 2, and 4. How does changing the value of h affect the graph?
3. Graph y = x2 + k for k = -4, -2, 2, and 4. Hoes does changing the value of k affect the graph?
Parabolas
Standard form of a quadratic function
Example 1: Sketch the graph of f(x) = x2 – 10x +25, and identify the vertex and axis of the parabola.
Example 2: Sketch the graph of f(x) = -x2 – 4x + 21, and identify the vertex and x intercepts
Example 3: Write the standard form of the equation of the parabola whose vertex is (-4,11) and that passes through the
point (-6,15)
Vertex of a parabola
2.2 Polynomial Functions of Higher Degrees
What are we going to learn?
-sketching polynomial functions
-determining the behavior of polynomial functions
Why are we learning it?
Polynomial functions can be used to model real world
situations.
What are some characteristics of a polynomial graph?
Exploration: Graphs of polynomials
For each function, identify the degree and state of the degree is even or odd. Identify the leading coefficient and state
whether it is positive or negative. Graph the function and describe the left-hand and right-hand behavior.
Function
degree
Even/
odd
Leading
coefficient
Positive or
negative
Left-hand
behavior
Righthand
behavior
f(x) = x3 – 2x2 – x + 1
f(x) = 2x5 + 2x2 – 5x + 1
f(x) = -2x5 – x2 + 5x + 3
f(x) = -x3 + 5x – 2
f(x) = 2x2 + 3x – 4
f(x) = x4 – 3x2 + 2x – 1
f(x) = x2 + 3x + 2
Leading coefficient test
1. When the degree is odd:
a. If the leading coefficient is positive
2. When the degree is even:
a. If the leading coefficient is positive
b. If the leading coefficient is negative
b. If the leading coefficient is negative
Example 1: Describe the left-hand and right-hand behavior of the graph of each function.
a. f(x) = ¼ x3 – 2x
b. f(x) = -3.6x5 + 5x3 – 1
Zeros of Polynomial Functions
Example 2: Find the zeros.
a. f(x) = -2x4 + 2x2
c. f(x) = x3 – 12x2 + 36x
b. f(x) = x2 – 25
d. f(x) = x2 + x -
Sketching the Graph of a Polynomial
1. Apply
2. Find
3. Plot
4. Draw
Example 3: Sketch f(x) = 3x4 – 4x3
Example 4: Sketch f(x) = 2x3 – 6x2
2.3 Polynomials and Division
What are we going to learn?
-use long division to divide polynomials
Example 1: Divide x3 – 2x – 9 by x – 3
Example 2: Divide x4 – 1 by x + 1
Example 3: Divide 6x4 – x3 – x2 + 9x – 3 by x2 + x - 1
Why are we learning it?
Long division can be used to factor more complicated
functions.
2.4 Complex Numbers
What are we going to learn?
-adding, subtracting, and multiplying complex numbers
-using complex conjugates
Why are we learning it?
Complex numbers are used to model some real life
problems.
What is i?
Example 1: Complete the following.
i1 =
i2 =
i3 =
i4 =
i5 =
i6 =
i7 =
i8 =
i9 =
i10 =
i11 =
i12 =
What pattern do you see?
How would you evaluate i raised to any positive power?
Complex Conjugates
Example 2: Multiple by its complex conjugate.
a. 3 + I
Example 3: Write the quotient
b. 5 – 4i
in standard form.
Example 4: Use the quadratic formula to solve.
a. 8x2 + 14x + 9 = 0
b. 7x2 + 5x + 2 = 0
2.5 Zeros of Polynomial Functions
What are we going to learn?
-finding zeros
-determining the number of zeros
Why are we learning it?
Zeros will be used in many future applications.
How many zeros does a polynomial have?
How do you find zeros?
Example 1: Find the zeros
a. y = x – 4
b. y = x2 – 6x + 9
c. y = x3 + 9x
Example 2: Find the zeros of f(x) = x3 – 15x2 + 75x – 125, given that x = 5 is a zero.
Example 3: Find the zeros of f(x) = 2x4 - 9x3 – 18x2 + 71x – 30, given that x = 2 is a zero.
d. y = x2 – 4x + 4
Complex zeros occur in conjugate pairs
Example 4: Find a third degree polynomial function with integer coefficients that has 2, 7i, and -7i as zeros.
Example 5: Find all the zeros of f(x) = x3 – 4x2 + 21x – 34, given that 1 + 4i is a zero of f.
Example 6: Write h(x) = -x3 + 11x2 + 41x – 51 as the product of linear factors, and list all its zeros.
2.6 Rational Functions
What are we going to learn?
-finding domains
-finding asymptotes
-analyzing and sketching graphs
Why are we learning it?
Rational functions are commonly used and it is
important to be able to analyze them.
What is a rational function?
Exploration: For each of the functions below:
a. Sketch a graph of the function.
b. What x value(s) are not in the domain of the function? (Where is it undefined?)
c. What happens to the curve as you approach those x values from the left?
d. What happens to the curve as you approach those x values from the right?
e. Does the function ever equal zero? If so, where?
1.
2.
3.
4.
5.
b.
b.
b.
b.
b.
c.
c.
c.
c.
c.
d.
d.
d.
d.
d.
e.
e.
e.
e.
e.
Horizontal Asymptotes
Vertical Asymptotes
Example 1: Find the asymptotes. Sketch the graph of the function.
a. ( )
b. ( )
c. ( )
d. ( )
e. ( )
f. ( )