1. Chapter 2 Polynomial and Rational Functions
Pre-Calculus
Chapter 2
Polynomial and Rational Functions

2. Warm Up 2.1
Find two positive real numbers whose product is a maximum and whose sum is 110.

3. 2.1 Quadratic Functions Objectives:
Analyze graphs of quadratic functions.
Write quadratic functions in standard form and use the results to sketch graphs of functions.
Find minimum and maximum values of functions in real-life applications.

4. Vocabulary Polynomial Function Standard Form of a Quadratic Function
Degree of Polynomial Function
Vertex Form of a Quadratic Function
Constant Function
Minimum Value of a Quadratic Function
Linear Function
Quadratic Function
Maximum Value of a Quadratic Function
Parabola
Axis of Symmetry
Vertex

5. Basic Functions Constant f (x) = a0 Linear f (x) = a1x + a0
Quadratic f (x) = a2x2 + a1x + a0
Cubic f (x) = a3x3 + a2x2 + a1x + a0
What is the next one?

6. Definition of a Polynomial Function
Polynomial Function of x with degree n
f (x) = anxn + an – 1 xn – 1 + … + a2x2 + a1x + a0
where:
n is a non-negative integer and
an, an – 1, … , a2, a1, a0 are real numbers
an ≠ 0

7. Quadratic Function Standard Form of a Quadratic Function:
f (x) = ax2 + bx + c
where a, b, and c are real numbers
and a ≠ 0
Used to model projectile motion such as rockets, baseballs, etc.

8. Graph of Quadratic Function
Let f (x) = ax2, a > 0
Shape of the graph?
Domain & range?
Increasing interval?
Decreasing interval?
Even or odd?
Vertex?

9. Graph of Quadratic Function
Let f (x) = ax2, a < 0
Shape of the graph?
Domain & range?
Increasing interval?
Decreasing interval?
Even or odd?
Vertex?

10. The Graphs

11. Example 1
Describe how the graph of each function is related to the graph of y = x2.
f (x) = 1/3 x2
g(x) = 2x2
h(x) = –x2 + 1
k(x) = (x + 2)2 – 3

12. Vertex Form of a Quadratic Function
Written as
f (x) = a(x – h)2 + k
where
the point (h, k) is the vertex of the parabola
the line x = h is the axis of symmetry.

13. Example 2
Describe the graph of f (x) = 2x2 + 8x + 7 and identify the vertex.
Rewrite the function in vertex form.
Hint: Complete the square.

14. Example 3
Describe the graph of f (x) = –x2 + 6x – 8 Identify the vertex and any x-intercepts.
Rewrite the function in vertex form to find the vertex.
Factor the original equation to find the x-intercepts.

15. Example 4
Write the equation of the parabola whose vertex is (1, 2) and that passes through the point (3, –6). Write the equation in both vertex form and standard form.

16. Minimum and Maximum Values
The minimum or maximum value of a quadratic function occurs at the ________.
If the equation is in vertex form, then the minimum or maximum value is ________.
If the equation is in standard form, then the vertex occurs at x = ___________. The min or max value is ___________.

17. Example 5 – Solve Algebraically and Graphically
A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45° with respect to the ground. The path of the baseball is given by the function f (x) = –0.0032x2 + x + 3, where f (x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

18. Example 6
A soft-drink manufacturer has daily production costs modeled by
C = 70,000 – 120x x2
where C is the total cost (in dollars) and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yield a minimum cost.

19. Example 7
The number h (in thousands) of hairdressers and cosmetologists in the U.S. can be approximated by the model
h = 4.17t2 – 48.1t + 881, 4 ≤ t ≤ 11
where t represents the year, with t = 4 corresponding to Determine the year in which the number of hairdressers and cosmetologists was the least.

20. Homework 2.1
Worksheet 2.1