foundations for functions chapter 1. 1-1 exploring functions terms you need to know –...

Chapter 1

Foundations for FunctionsChapter 1 1-1 Exploring FunctionsTerms you need to know Transformation, Translation, Reflection, Stretch, and Compression

Perform the transformations to the point (-1,2). What is the new coordinate?

1a. 3 units left

b. 4 units right, 3 units up

Use a table to transform

2a. 2 units up b. Across x-axis

4

Use a table to perform a 3a. vertical stretch of factor 2

b. horizontal compression of factor .1-2 Introduction to Parent FunctionsTerms you need to know Parent Function, Linear, Quadratic, Cubic, and Square root

Identify the parent function and graph on the calculator. Describe the transformation. State Domain and Range

Given the following points, graph the data and identify the parent function. Describe the transformation that bests fits the data.

Graph the relationship from year to sales in millions of dollars. Which parent function best describes it? Use graph to estimate when sales reach $10 million. YearSales (million$)1.621.834.247.8512.6

1-3 Transforming Linear FunctionsLet g(x) be the indicated transformation of f(x). Write the rule for g(x).

x-202f(x)0122.3. Let g(x) be a horizontal compression of f(x) = - x+4 by a factor of . Write the rule and graph it.

4. Let g(x) be a horizontal shift of f(x) = 3x left 6 units followed by a horizontal stretch of factor 4. Write the rule for g(x)The golf team is selling T-shirts for a fundraiser. The function R(n) = 7.5n represents the teams revenue in dollars and the number of shirts sold (n). a. The team paid $60 for the shirts. Write a function P(n) for the teams profit.b. Graph both functions and describe the transformation applied.

1-4 Curve fitting with Linear modelsTerms you need to know regression, correlation, line of best fit, correlation coefficientPositive correlation y tends to __________ as x __________.

Negative correlation y tends to __________ as x __________.

Relatively no correlation no ________ pattern.

x-1-0.50.511.5233.544.2y8875.510330.50-2Draw the scatterplot. Name the type of correlation, sketch the line of best fit, and find its equation.

Example: Constructing a Scatter PlotA marketing manager conducted a study to determine whether there is a linear relationship between money spent on advertising and company sales. The data are shown in the table. Display the data in a scatter plot, determine correlation, find the correlation coefficient, and the line of best fit using the calculator. Do a screen sketch.

Advertisingexpenses,($1000), xCompanysales($1000), y2.42251.61842.02202.62401.41801.61842.01862.2215Solution: Constructing a Scatter Plot Using TechnologyEnter the x-values into list L1 and the y-values into list L2.Use Stat Plot to construct the scatter plot.STAT > Edit

STATPLOTFrom the scatter plot, it appears that the variables have a____________linear correlation.Predict sales if $2500 was spent in advertising.

T eac h i ng T ra ns pa r e ncy

Copyright by Holt McDougal. Holt McDougal Algebra 2All rights reserved.

Exploring Transformations

TranslationsHorizontal Translation Vertical Translation

Each point shifts right or left by a number of units.

The x-coordinate changes.

(1, 2) (1 + 3, 2) (x, y) (x + h, y)

left if h < 0 right if h > 0

Each point shifts up or down by a number of units.

The y-coordinate changes.

(1, 2) (1, 2 + 2 ) (x, y) (x, y + k )

down if k < 0 up if k > 0

y

x

4

2 (4, 2) (1, 2)

3 units

0 2 4

y

x

4

2 (1, 2)

(1, 4)

2 units

0 2 4

ReflectionsReflection Across y-axis Reflection Across x-axis

Each point flips across the y-axis.


(1, 2) (-1, 2) (x, y) (-x, y)

Each point flips across the x-axis.


(1, 2) (1, -2 ) (x, y) (x, -y ) 2 0

y

x 1 unit 1 unit

-2

(-1, 2) (1, 2)

0

y

x

(1, -2)

(1, 2)

2 units

2 units

2

-2

2

1 - 1

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Exploring Transformations (continued)

Stretches and CompressionsHorizontal Vertical

Stretch Each point is pulled away from the y-axis.


(4, 0) (2 (4) , 0) (x, y) (bx, y) b > 1

Each point is pulled away from the x-axis.


(0, 4) (0, 2 (4) ) (x, y) (x, ay )

a > 1

Compression Each point is pushed toward the y-axis.


(4, 0) ( 1 _ 2 4 , 0) (x, y) (bx, y) 0 < b < 1

Each point is pushed toward the x-axis.


(0, 4) (0, 1 _ 2 (4) ) (x, y) (x, ay )

0 < a < 1

3

1 - 1

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Introduction to Parent Functions

Parent Functions

Family Constant Linear Quadratic CubicSquare

root

Rule f (x) = c f (x) = x f (x) = x 2 f (x) = x 3 f (x) = x

Graph

Domain x 0

Range y = c y 0 y 0

Intersects y-axis

(0, c) (0, 0) (0, 0) (0, 0) (0, 0)

y

x

f

y

x

f

y

x

f

y

x

f

Linear Quadratic Cubic Square Rootf x x f x x 2 f x x 3 f x x

6

1 - 2




exploration


Although there are an infinite number of functions, many functions can be grouped into families that have similar characteristics. 1. Use your calculator to graph y 5 x 2 by pressing Y= ,

entering Y1 5 X 2 , and pressing GRAPH . Describe the graph.

2. Enter and graph Y2 5 X 2 1 2. Describe the graph of Y2 as compared to the graph of Y1.

3. Enter and graph Y3 5 2X 2 . Compare this graph to the graphs from Problems 1 and 2.

4. Enter and graph Y4 5 2X 2 2 6. Compare this graph to the graphs above.

5. Make a conjecture about the shape of the graphs of functions of the form y 5 ax 2 1 b.

Think and discuss 6. Explain what the graphs of functions of the form y 5 mx 1 b

look like.

Copyright by Holt, Rinehart and Winston. 2 Holt McDougal Algebra 2All rights reserved.

x-x


1-2

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exploration


Although there are an infinite number of functions, many functions can be grouped into families that have similar characteristics. 1. Use your calculator to graph y 5 x 2 by pressing Y= ,

entering Y1 5 X 2 , and pressing GRAPH . Describe the graph.

2. Enter and graph Y2 5 X 2 1 2. Describe the graph of Y2 as compared to the graph of Y1.

3. Enter and graph Y3 5 2X 2 . Compare this graph to the graphs from Problems 1 and 2.

4. Enter and graph Y4 5 2X 2 2 6. Compare this graph to the graphs above.

5. Make a conjecture about the shape of the graphs of functions of the form y 5 ax 2 1 b.

Think and discuss 6. Explain what the graphs of functions of the form y 5 mx 1 b

look like. 1. Possible answer: The graph is U shaped, with the lowest point at 0, 0 .

2. The graph of Y2 is the graph of Y1 shifted 2 units up.

3. The graph of Y3 is narrower than the graphs of Y1 and Y2, with its lowest point at 0, 0 . 4. The graph of Y4 is narrower than the graphs of Y1 and Y2, with its lowest point

at 0, 26 . 5. Possible answer: The graphs are U shaped, with the lowest point at 0, b . The value of

a determines how narrow the graphs are.

6. Possible answer: The graphs are lines that cross the y-axis at 0, b . The value of m determines how steep the lines are.


x-x


1-2

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Curve Fitting with Linear Models

Positive correlation, positive slope

Negative correlation, negative slope

Relatively no correlation

r is a value in the range -1 r 1.

If r = 1, the data set forms a straight line with a positive slope.

If r = 0, the data set has no correlation.

If r = -1, the data set forms a straight line with a negative slope.

r -0.95 r -0.6 r 0 r 0.6 r 0.95

Properties of the Correlation Coefficient r

13

1 - 4




foundations for functions chapter 1. 1-1 exploring functions terms you need to know –...

Documents

linear correlation

y x h

negative correlation

type of correlation

y left

number of units

xcoordinate changes

use stat plot