2.1a Polynomial Functions2.1a Polynomial FunctionsLinear FunctionsLinear Functions

Linear Correlation/ModelingLinear Correlation/Modeling

Lots of new/old info. toLots of new/old info. to

start our next chapter!!!start our next chapter!!!

Polynomial Functions – What are they?

Definition: Polynomial Function

Let n be a nonnegative integer and letbe real numbers with . The function given by

0 1 2 1, , , , ,n na a a a a0na

1 21 2 1 0

n nn nf x a x a x a x a x a

nais a polynomial function of degree n. The leadingcoefficient is .

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

0f x

Identifying Polynomial Functions

Which of the following are polynomial functions? For those thatare polynomial functions, state the degree and leading coefficient.For those that are not, explain why not.

3 14 5

2f x x x A polynomial function!!!A polynomial function!!!

46 7g x x

Degree 3, leading coeff. 4Degree 3, leading coeff. 4

Not a polynomial function!!!Not a polynomial function!!!

The exponent – 4 messesThe exponent – 4 messeseverything up!!!everything up!!!

Identifying Polynomial Functions

Which of the following are polynomial functions? For those thatare polynomial functions, state the degree and leading coefficient.For those that are not, explain why not.

4 29 16h x x x Not a polynomial function!!!Not a polynomial function!!!

415 2k x x x

Cannot be simplified toCannot be simplified topolynomial form!!!polynomial form!!!

A polynomial function!!!A polynomial function!!!

Degree 4, leading coeff. –2Degree 4, leading coeff. –2

Can we simplify this one?Can we simplify this one?

Polynomial Functions of No and LowDegree (some familiar ones?)Name

0f x Form Degree

Zero Function Undefined

f x aConstant Function 0 0a

f x ax b Linear Function 1 0a

2f x ax bx c Quadratic Function 2

0a

Today we’ll focus mainly on this one…Today we’ll focus mainly on this one…

Linear Function: f (x) = ax + b(where a and b are constants, and

a is not equal to zero)

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

A line in the Cartesian plane is the graph of a linear function ifand only if it is a slant line, that is, neither horizontal or vertical.

This is the “Do Now”!!!This is the “Do Now”!!!

ff (x) = –x + 1 (x) = –x + 1

Points: 1,2 , 3, 2 Slope:2 2

13 1

m

Point-Slope Form:

2 1 1y x

Average Rate of Change

The average rate of change of a function betweenx = a and x = b, if a and b are not the same, is

y f x

f b f a

b a

Constant Rate of Change. A function defined on all realnumbers is a linear function if and only if it has a constantnonzero average rate of change between any two points onits graph.

SLOPE!!!SLOPE!!!Does this equationDoes this equation

look familiar???look familiar???

The slope The slope mm in the formula in the formula ff ( (xx) = ) = mxmx + + bb is the rate of is the rate of change of the linear function.change of the linear function.

Characterizing the Nature of a LinearFunction

Point of View

0m f x mx b

Characterization

Verbal Polynomial of degree 1

Algebraic

Graphical Slant line with slope m and y-intercept b

Analytical Function with constant nonzero rate ofchange m: f is increasing if m > 0,decreasing if m < 0; initial value of thefunction = f (0) = b

Linear Correlation and Modeling

When the points of a scatter plot are clustered along a line, wesay there is a linear correlation between the quantitiesrepresented by the data.

Positive correlation – if the “cluster line” has positive slope

Negative correlation – if the “cluster line” has negative slope

(Linear) correlation coefficient (r) – measures the strengthand direction of the linear correlation of a data set

Properties of the Correlation Coefficient

1. 1 r 1 2. When r > 0, there is a positive linear correlation

3. When r < 0, there is a negative linear correlation

4. When |r| is near one, there is a strong correlation

5. When r is near zero, there is weak or no linear correlation

Estimating Linear Correlations

Strong positivelinear correlation

Weak positivelinear correlation

Estimating Linear Correlations

Strong negativelinear correlation

Weak negativelinear correlation

Estimating Linear Correlations

Little or nolinear correlation

Let’s Try a Regression Problem!

Steps for Regression Analysis

1. Enter and plot the data (scatter plot).

2. Find the regression model that fits the problem situation.

3. Superimpose the graph of the regression model on the scatter plot, and observe the fit.

4. Use the regression model to make the predictions called for in the problem.

Let’s Try a Regression Problem!

Weekly Sales Data forBoxes of Cereal

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

Use these data to write a linearmodel for demand (in boxessold per week) as a functionof the price per box (in dollars).Describe the strength anddirection of the linear correlation.Then use the model to predictweekly cereal sales if the priceis dropped to $2.00 or raised to$4.00 per box.

Let’s Try a Regression Problem!

Weekly Sales Data forBoxes of Cereal

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

Enter the data (price in L1, boxessold in L2) and draw a scatter plot.

Use your calculator to obtain alinear regression equation:

15,358.929 73,622.5y x The correlation coefficient indicatesa strong negative linear correlation:

0.981r

Let’s Try a Regression Problem!

Weekly Sales Data forBoxes of Cereal

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

Graph the regression line on top ofthe scatter plot, and then calculatey at x = 2.00 and 4.00:

At a price of $2.00, approximatelyAt a price of $2.00, approximately42,905 boxes of cereal would be42,905 boxes of cereal would besold. At $4.00, the number soldsold. At $4.00, the number solddrops to approximately 12,187.drops to approximately 12,187.

Homework: p. 175-177 1-17 oddHomework: p. 175-177 1-17 odd