Pre-CalculusChapter 2

Polynomial and Rational Functions

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Warm Up 2.1

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

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2.1 Quadratic FunctionsObjectives:

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.

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Vocabulary Polynomial

Function Degree of

Polynomial Function

Constant Function Linear Function Quadratic Function Parabola Axis of Symmetry

Vertex Standard Form of a

Quadratic Function Vertex Form of a

Quadratic Function Minimum Value of a

Quadratic Function Maximum Value of

a Quadratic Function

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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?

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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 numbersan ≠ 0

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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.

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Graph of Quadratic Function Let f (x) = ax2, a > 0

Shape of the graph?Domain & range?Increasing interval?Decreasing interval?Even or odd?Vertex?

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Graph of Quadratic Function Let f (x) = ax2, a < 0

Shape of the graph?Domain & range?Increasing interval?Decreasing interval?Even or odd?Vertex?

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The Graphs

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Example 1 Describe how the graph of each

function is related to the graph of y = x2.

a. f (x) = 1/3 x2

b. g(x) = 2x2

c. h(x) = –x2 + 1

d. k(x) = (x + 2)2 – 3

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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.

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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.

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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.

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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.

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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 ___________.

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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?

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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.

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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.

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Homework 2.1 Worksheet

2.1