2-1 Relations and Functions

Relation – Function – Domain – Range – State the Domain and Range for each relation. Then determine if each relation is a function.

1) {(2, 3), (2, 4), (2, 5)} 2) {(3, 1), (4, 2), (5, 3)} 3)

4) 5) D R D R

One-to-One Function Onto Function Both One-to-One and Onto

Each element of the domain

pairs to exactly one unique

element of the range.

Each element of the range also

corresponds to an element of the

domain.

Each element of the domain is paired

to exactly one element of the range

and each element of the range.

x y

0 -2

3 2

6 2

2

A B C

2

A B C

List the domain and range for each relation. Then determine whether the relation is a function. If it is a function, it is one-to-one, onto, both or neither. 6) {(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)} 7)

8)

9) Map the following. {(-4,-1), (4,0), (0,3), (2,0)}. Is it a function? ________ Is it one-to-one, onto, neither or both? ________

Vertical Line Test – Which of the following are functions?

(0, -2) (-4, -2)

(-3, 1) (1, 2)

(3, 3)

x 1 2 3

y 2 4 6

Discrete relation – Continuous relation –

10) Graph 12 xy .

Is it a function? ________ Is it one-to-one, onto or both? ________ Discrete or continuous? ________ Find the domain and range. D: _____________ R: _____________ 11) REAL WORLD: Cool Athletics introduced the new Power Sneaker in one of their stores. The table shows the sales for the first 6 weeks.

a. Graph the data. b. Identify the domain and range. c. Is the relation discrete or continuous?

Functional notation – f(x) – said as “f of x ” Independent variable – Dependent variable –

Find each value if 2

5)(

xxf and 32)( xxg .

3) (3)f 4) ( 4)f 5) ( 6)g

6) g1

2

7) ( )f a 8) ( 2)g m

Week 1 2 3 4 5 6

Pairs Sold 8 10 15 22 31 44

2-2 Linear Relations and Functions

Linear Relation - Linear equation - Linear Equations NOT Linear Equations

Linear Function – State whether each function is a linear function. Write yes or no. Explain.

1) ( ) 5 2f x x 2) ( ) 2g x x 3) 2( ) 2h x x 4)

2( )p x

x 5) ( ) 3 18q x xy

REAL WORLD: Meteorology

The linear function ( ) 1.8 32f C C can be used to find the number of degrees Fahrenheit f(C)

that are equivalent to a given number of degrees Celsius C. a. On the Celsius scale, normal body temperature is 37° C. What is it in degrees Fahrenheit? b. There are 100 Celsius degrees between freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree?

Standard Form - Ax + By = C, where A, B, and C are integers whose greatest common factor is 1. A≥0, and A and B are both nonzero.

Example: 3x + 5y = 12; A = 3, B = 5, and C = 12. Write each equation in standard form. Identify A, B, and C.

6) 3 9y x 7) 4 5 7x y 8) 8 4 12y x 9) 1

23y x

y-intercept - x-intercept - Use intercepts to graph a line: Find the x-intercept and y-intercept of each and then

graph the equation.

10) 2 5 10x y 11) 2 4 0x y

(0, ___) (0, ___) (____, 0) (____, 0) Write each function in slope-intercept form. Then graph the function.

12) 4 1x y 13) 3

64

x y

Horizontal Lines – Vertical Lines – Determine if each line is horizontal or vertical. Then graph. 14) x = 2 15) y = -4

2-3 Rate of Change and Slope

Rate of Change – REAL WORLD 1) 2)

Slope – Finding Slope: Find the slope of the line that passes through each pair of points.

3) (-1, 1), (2, -5) 4) (-1, 4), (1, -2) 5) (-2, -8), (7, -8) Determine the rate of change for each graph.

6) 7)

Year Number of

Banks

1993 3304

1995 2858

1997 2597

1999 2363

College Admissions In 2004, 56,878 students applied to UCLA. In 2006, 60,291 students applied. Find the rate of change in the number of students applying for admission from 2004 to 2006.

Business The table shows the number of national banks that were operating in the U.S. in various years. Find the rate of change from 1993 to 1999.

2-4 Writing Linear Equations Slope-intercept form – Write the equation in slope-intercept form for each graph. 1) 2) 3)

_________________ _________________ __________________ Point-slope form – Given point and slope: Write the equation in slope-intercept for each line described.

4) passes through (-4,6); 2m 5) passes through (5,-2); 3

5m

Given two points: Write an equation of a line passing through each pair of points.

6) (-1, 1), (2, -5) 7) (-2, -8), (7, -8) 8) (5, 4), (3, 7)

Slope of Parallel Lines – Slope of Perpendicular Lines –

Write the equation of each line in slope intercept form. 9) 10) 11) 12)

Parallel to a line with the equation

3 6x y and passes through: (2,3)

Perpendicular to a line with the equation

4 3y x and passes through: (-8, -7)

Perpendicular to a line with the

equation 3

12

y x and passes

through: (9, -2)

Parallel to a line with the equation 1.8 7y x and passes through: (5, 2)

2-5 Scatter Plots and Lines of Regression

Bivariate data -

Scatter plot (dot plot) –

Line of fit – Prediction equation –

Negative correlation,

Negative slope

REAL WORLD: 1) Sports Use the table with player minutes played compared to points scored. a. Make a scatter plot and line of fit with minutes played as the independent variable. b. Describe the type of correlation. c. Use two ordered pairs to write a prediction equation. d. Predict the points scored, if a player were to play for 24 minutes. e. How accurate does your prediction appear to be?

Regression line –

Directions for Graphing Calculator

Enter the data into lists L1 and L2 on a graphing calculator.

29 30 …

Set viewing window to fit data. 25 75 5

70 110

Graph scatter plot. [STAT PLOT]

Find the regression equation. 4

Graph the regression equation. 5 1

Predict using the function. [CALC] 1 86

2) Find the following for this data on average temperature and rainfall for eight months in Boston, MA. a. Use a graphing calculator to make a scatter plot of the data. b. Find an equation for and graph a line of regression. c. Predict the temperature when the rainfall is 86 mm. d. Identify the correlation coefficient.

STAT CALC

ENTER

Y= VARS

STAT

GRAPH

2nd ENTER

ENTER ENTER

WINDOW ENTER ENT ER ENTER

ENTER ENTER

2nd ENTER ENTER GRAPH

ENTER

2-6 Special Functions

Piecewise-defined functions – Graph each piecewise function.

1) 4 if 2

( ) 1 if 2

x xf x

x

2)

2 if 3

( ) 2 if -3 1

2 2 if 1

x x

f x x

x x

Write the piecewise defined function shown in the graph. 3) 4)

( )f x

( )f x

Step function – Greatest integer function –

Absolute-value function -

Vertex of an Absolute-Value Function – (h, k) in Where “h” is the horizontal movement and k is the vertical movement.

Graph each function. Identify the domain and range.

6) ( ) 1f x x 7) ( ) 2f x x

8) ( ) 3f x x 9) ( ) 3f x x

x f(x)

x f(x)

x f(x)

x f(x)

REAL WORLD: 5) Psychology One psychologist charges for counseling sessions at the rate of $90 per hour or any fraction thereof. Draw a graph that represents this situation.

2-7 Parent Functions and Transformations

Family of Graphs –

Parent Function – Indentify the type of function represented by each graph.

1) _____________ 2) _____________ 3) _____________ 4) ____________

Translation – Describe each translation. Then graph the function.

5) 2y x 6) 3y x 7) 2 1y x

_____________ _____________ _____________ _____________ _____________ _____________

Reflection - Describe each reflection. Then graph the function.

8) y x 9) y x 10) 2y x

_____________ _____________ _____________ _____________ _____________ _____________

Dilation – Describe each dilation. Then graph the function.

11) 3y x 12) 1

2y x 13)

22y x

_____________ _____________ _____________ _____________ _____________ _____________

14) REAL WORLD The function 21

( ) 3 4.52

f x x can be used to represent a parabolic

archway. Describe the transformations in the function. Then graph the function.

_______________________________________ _______________________________________

_______________________________________

2-8 Linear and Absolute Value Inequalities Linear Inequality -

Boundary Line : dashed - solid - Graph each inequality.

1)

2)

3) 4) 3x Graph each inequality using x and y intercepts.

5) 6)

Graph each absolute value inequality.

8) ( ) 4f x x 9) ( ) 2f x x

REAL WORLD: 7) A school carnival charges $4.50 for adults and $3.00 for children. The school needs to make at least $135 to cover expenses.

a) Using x as the number of adult tickets sold and y as the number of child tickets sold, write and graph an inequality for the amount the school makes on ticket sales.

b) If 25 child tickets are sold, how many adult tickets must be sold to cover expenses?