Linear Relations and Functions

DOMAIN: The set of x coordinates from a group of ordered pairs

RANGE: The set of y coordinates from a group of ordered pairs

FUNCTION: a type of relation in which each element of the domain is mapped with EXACTLY one element of the range

ONE-TO-ONE FUNCTION: each element of the range is paired with exactly one element of the domain

DISCRETE: a relation in which the domain is a set of individual points.

CONTINUOUS: a relation with an infinite number of elements and can be graphed continuously as a line or smooth graph.

VERTICAL LINE TEST: used to determine if a relation is a function

2.1

Domain: {-4, -3, 0, 1, 3}

Range:{-2, 0, 1, 2, 3}

It is a function(-4,0)

(-3,1)

(0,-2)

(1, 2)

(3,3)

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

-1

1

4

3

5

NOTE this is a function, each x is mapped to exactly one y

{(5,6) (-3,0) (1,1) (-3,6)}

Graph y=3x-1 then find the domain and range, determine if it is discrete or continuous

Graph y=x2 + 1 and find the domain and range. Determine if it is discrete or continuous

Given f(x)= x3 – 3◦ Find f(1)

◦ Find f(-2)

◦ Find f(2y)

Linear Functionf(x)=mx + b

*Have a highest exponent of 1

Linear Equationy=mx+b

*Have a highest exponent of 1

2.2

1. State whether each function is a linear function, explain.◦ g(x)=2x-5

g(x) is a linear function because the highest exponent in 1 and it is in slope intercept form m=2 and b = -5

◦ p(x)=x3+2 p(x) is not a linear function because x has an

exponent > 1

◦ f(x)= 4+7x f(x) is a linear function because the highest exponent

is 1 and it can be written in slope intercept form with m=7 and b = 4

Write each equation in standard form. Identify A, B, and C

123

2.2 yx93.1 xy

Graph the equation by the intercepts.◦ Find the x-int and y-int by substituting the other

letter with a zero (write as ordered pairs)

-2x + y – 4 = 0

12

12

xx

yym

2.3

Positive slope Negative Slope Zero Slope Undefined Slope

Parallel Lines have the same slope Perpendicular lines have slopes that are

opposite signs and reciprocals

A. (1, -3) (3, 5)

B. A line parallel to x – 3y = 3

C. A line perpendicular to (2, 2) (4, 2)

Passes through (2, -5) parallel to the graph of x = 4

Passes through the origin perpendicular to the graph of y = -x

Slope-Intercept Form: y = mx + b m is slope and b is the y-intercept

Point-Slope Form: y – y1 = m (x – x1) m is slope and y1 and x1 are any ordered pair on the

line

2.4

A. Through (6, 1) and (8, -4)

B. Through (-5, 7) perpendicular to y = ½x + 6

2.5

Graph Ordered Pairs Select two points to connect for the line of

best fit. Write equation of that line using those two

points to find slope Answer any additional questions using the

equation you just wrote.

EDUCATION The table below shows the approximate percent of students who sent applications to two colleges in various years since 1985. Make a scatter plot of the data and draw a line of fit.

Graph the data as ordered pairs, with the number of years since 1985 on the horizontal axis and the percentage on the vertical axis.

The points (3, 18) and (15, 13) appear to represent the data well. Draw a line through these two points

Find a prediction equation. What do the slope and y-intercept indicate?

Find an equation of the line through (3, 18) and (15, 13). Begin by finding the slope

Slope formula

Substitute.

Simplify.

Simplify.

Distribute.

Substitute.

Point-slope form

Answer: One prediction equation is y = –0.42x + 19.26.

The slope indicates that the percent of students sending applications to two colleges is falling about 0.4% each year. The y-intercept indicates that the percent in 1985 should have been about 19%.

Predict the percent in 2010 The year 2010 is 25 years after 1985, so use the

prediction equation to find the value of y when x = 25.

Answer: The model predicts that the percent in 2010 should be about 9%.

Simplify.

x = 25

Prediction equation

Graphing Inequalities

• The equation makes the line to define the boundary

• The shaded region is the half-plane

1. Get the equation into slope-intercept form2. Graph the intercept and use the slope to find at

least 2 more points3. Draw the line (dotted or solid)4. Test an ordered-pair not on the line

1. If it is true shade that side of the line2. If it is false shade the other side of the line

2.7

Ex: 3y - 2 > -x + 7

3

1

3y – 2 > -x + 7 +2 +2

3y > -x + 9/3 /3 /3

y > - x + 3

m = -

b = 3 = (0, 3)3

1

Test: (0, 0)

0 > - (0) + 3

0 > 0 + 3

0 > 3 false (shade other side)

3

1

< or > or

Dotted Line Solid Line