PRE-AP PRE-CALCULUSCHAPTER 2, SECTION 1Linear and Quadratic Functions and Modeling

2013 - 2014

Polynomial Functions• Functions with…

• Whole number exponents• No radicals around the variable• No variables in the denominator

• The zero function is a polynomial function. It has no degree and no leading coefficient.

• Polynomial functions are defined and continuous on all real numbers.

Polynomial Functions• If you determine a function to be a polynomial, you can

determine the degree of the function and the leading coefficient.

Which of the following are polynomial functions?If they are, state the degree and leading coefficient.

𝑓 (𝑥 )=4 𝑥2−5 𝑥− 12

𝑔 (𝑥 )=6 𝑥− 4+7

h (𝑥 )=√9 𝑥4+16 𝑥2

𝑘 (𝑥 )=15𝑥−2𝑥4

Polynomial Functions of No and Low Degree

Name Form Degree

Zero function Undefined

Constant Function

0

Linear Function 1

Quadratic Function

2

Provide Examples of the following:

• Zero Function:

• Constant Function:

• Linear Function:

• Quadratic Function:

Linear Function

• A linear function is a polynomial function of degree 1.• ________________ are not graphs of functions because

they fail the vertical line test. • ________________ are graphs of constant functions.• A line is the graph of a _______________ if and only if it

is a slant line.

Average Rate of Change• If a function is between x=a and x=b, where a≠b, then the

rate of change can be found by .

Camelot Apartments bought a $50,000 building and for tax purposes are depreciating it $2000 per year over a 25-yr period using straight-line depreciation.

What is the rate of change of the value of the building?

Write an expression for the value v(t) of the building as a linear function of the time t since the building was placed in service.

Evaluate v(0) and v(16).

Solve v(t) = 39,000

Linear Correlation and Modeling

R is the correlation coefficient.

Use the data in the table to write a linear model for demand (in the boxes sold per week) as a function of the price per box (in dollars).

Weekly Sales Data Based on Marketing Research

Price Per box Boxes Sold

$2.40 38,320

$2.60 33,710

$2.80 28,280

$3.00 26,550

$3.20 25,530

$3.40 22,170

$3.60 18,260

What are the steps for plotting a scatterplot in the calculator, then finding the equation of the line?

Quadratic Functions & Their Graphs• A quadratic function is a polynomial function of degree

____. • The graph of the squaring function _________ is a

_____________. • The parent function of any upward- or downward-opening

parabola is ___________.

Describe the transformation of into the graphs of the given equations. Sketch the graphs by hand.

𝑔 (𝑥 )=− 12𝑥2+3h (𝑥 )=3 (𝑥+2)2−1

Standard Quadratic Formula

• This form is helpful when factoring or using the _______________________ to find the zeros of the functions.

Find the zeros of the function.

𝑓 (𝑥 )=−3 𝑥2+6 𝑥−5

Find the roots of the function

−2 𝑥2+7𝑥−3

Comparing Standard Form & Vertex Form

• If you expand the vertex form , we can obtain the formulas for h and k.

Vertex Form• Any quadratic function , can be written in the

____________.

• The graph of the function has the vertex (h, k)• h=_________• k= _________• If , the parabola opens _________• If , the parabola opens __________

Use the vertex form of a quadratic function to find the vertex and axis of the graph of . Rewrite the equation in vertex form.

Use the vertex form of a quadratic function to find the vertex and axis of the graph of . Rewrite the equation in vertex form.

The nature of a Quadratic functionPoint of View Characterization

Verbal Polynomial of degree 2

Algebraic

Graphical Parabola with vertex (___,___) and axis x=___Opens upwards if a 0Opens downwards if a 0Initial value = y-intercept = = c

x-intercepts =

Application of Quadratics• Julie Stone designed a rectangular patio that is 25 ft by 40

ft. This patio is surrounded by a terraced strip of uniform with small trees and shrubs. If the are A of this terraced strip is 504 square feet, find the width x of the strip.

Application of Quadratics• The per unit price p (in dollars) of a popular toy when x

units (in thousands) are produced is modeled by the function .

• • The revenue (in thousands of dollars) is the product of the

price per unit and the number of units (in thousands) produced. That is,

• How many units should be produced if the total revenue is to be $1,000,000?

Vertical Free Fall of Motion• The height s and vertical velocity v of an object in free fall

are given by

Where t is time (in seconds), is the acceleration due to gravity, is the initial vertical velocity of the object, and is its initial height.

Vertical Free Fall Application• As a promotion for the Houston Astros downtown ballpark,

a competition is held to see who can throw a baseball the highest from the front row of the upper deck of seats, 83 ft above field level. The winner throws the ball with an initial vertical velocity of 92 ft/sect and it lands on the infield grass.

a) Find the maximum height of the baseball.

b) How much time is the ball in the air?

c) Determine its vertical velocity when it hits the ground.

Critical Thinking:

Write an equation for the linear function f such that f(-1)=2 and f(3)=-2.

Critical Thinking:

Write an equation for the linear function f such that f(0)=3 and f(3)=0.

Homework for Chapter 2.1

• Page 182 - 187: #’s 2, 9, 12, 13, 16, 22, 24, 25, 27, 40, 45, 47, 66, 75, 78 (a & b)

• (total of 15 problems)