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

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

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.

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

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?

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

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.

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

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

The Graphs

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

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.

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.

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.

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.

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 ___________.

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?

Example 6 A soft-drink manufacturer has daily production costs modeled by C = 70,000 – 120x + 0.055x2 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.

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 1994. Determine the year in which the number of hairdressers and cosmetologists was the least.

Homework 2.1 Worksheet 2.1