2.7 – Absolute Value Inequalities To solve an absolute value inequality, treat it as an equation. Create two inequalities, where the expression in the

2.7 – Absolute Value InequalitiesTo solve an absolute value inequality, treat it as an equation.

843 x

123 x

843 x

4x

843 x

843 x

43 x

3

4x

4,

3

4

Create two inequalities, where the expression in the absolute value is set to be positive and negative.

2.7 – Absolute Value InequalitiesAlternate Method

843 x

4,

3

4

If a is a positive number, then X < a is equivalent to a < x < a.

2.7 – Absolute Value InequalitiesExample

Solve x + 4 < 6

6 < x + 4 < 6

6 – 4 < x + 4 – 4 < 6 – 4

10 < x < 2

(10, 2)

2.7 – Absolute Value Inequalities

Solve x 3 + 6 7

x 3 1

1 x 3 1

2 x 4

[2, 4]

x 3 + 6 - 6 7 - 6

1 + 3 x 3 + 3 1 + 3

Example


Solve 8x 3 < 2No solution

An absolute value cannot be less than a negative number, since it can’t be negative.

Example

2.7 – Absolute Value InequalitiesIf a is a positive number, then X > a is equivalent to

X < a or X > a.

843 x

123 x

843 x

4x

843 x

843 x

43 x

3

4x

or

,4

3

4,


Solve 10 + 3x + 1 > 2

Example

10 + 3x > 1

10 + 3x < 1

3x < 11 3x > 9

(, ) (3, )3

11

or 10 + 3x > 1

2.7 – Absolute Value InequalitiesExample

42

7

xSolve

1[]

-15

or

LCD: 2

Defn: A relation is a set of ordered pairs.

2,4,2,4,1,1,1,1,0,0 A

:Arange

4,1,0

2,1,0,1,2

Domain: The values of the 1st component of the ordered pair.

:Adomain

Range: The values of the 2nd component of the ordered pair.

3.2 – Introduction to Functions


:range 6,5,4,3

3,2,1,4:domain

x y

1 3

2 5

-4 6

1 4

3 3

x y

4 2

-3 8

6 1

-1 9

5 6

x y

2 3

5 7

3 8

-2 -5

8 7

:range 9,8,6,2,1

6,5,4,1,3 :domain

:range 8,7,3,5

8,5,3,2,2:domain

State the domain and range of each relation.

The Rectangular Coordinate System

Ordered Pair

(x, y)

(independent variable, dependent variable)

(1st component, 2nd component)

(input, output)

(abscissa, ordinate)


x

y

1st Quadrant2nd Quadrant

3rd Quadrant 4th Quadrant

The Rectangular Coordinate System

,x y ,x y

,x y ,x y


Defn: A function is a relation where every x value has one and only one value of y assigned to it.

x y

1 3

2 5

-4 6

1 4

3 3

x y

4 2

-3 8

6 1

-1 9

5 6

x y

2 3

5 7

3 8

-2 -5

8 7

function not a function function

State whether or not the following relations could be a function or not.


Functions and Equations.

32 xy

x y

0 -3

5 7

-2 -7

4 5

3 3

x y

2 4

-2 4

-4 16

3 9

-3 9

x y

1 1

1 -1

4 2

4 -2

0 0

2xy 2yx

function function not a function


State whether or not the following equations are functions or not.


Vertical Line Test

Graphs can be used to determine if a relation is a function.

If a vertical line can be drawn so that it intersects a graph of an equation more than once, then the equation is not a function.

x

y

The Vertical Line Test

32 xy

x y

0 -3

5 7

-2 -7

4 5

3 3


function

x

y

2xy x y

2 4

-2 4

-4 16

3 9

-3 9



function

x

y

2yx x y

1 1

1 -1

4 2

4 -2

0 0



not a function

Find the domain and range of the function graphed to the right. Use interval notation.

x

y

Domain:

Domain

Range:

Range[–3, 4]

[–4, 2]


Domain and Range from Graphs

Find the domain and range of the function graphed to the right. Use interval notation.

x

y

Domain:

DomainRange:

Range

(– , )

[– 2, )


Domain and Range from Graphs


Function Notation

13 xxf 13 xxy

xfxyy

Shorthand for stating that an equation is a function.

13 xy

Defines the independent variable (usually x) and the dependent variable (usually y).

52 xxf

Function notation also defines the value of x that is to be use to calculate the corresponding value of y.

5323 f

13 f

1,3


f(x) = 4x – 1find f(2).

f(2) = 4(2) – 1

f(2) = 8 – 1

f(2) = 7

(2, 7)

g(x) = x2 – 2xfind g(–3).

g(–3) = (-3)2 – 2(-3)

g(–3) = 9 + 6

g(–3) = 15

(–3, 15)

find f(3).

Given the graph of the following function, find each function value by inspecting the graph.

f(5) = 7x

y

f(x)

f(4) = 3

f(5) = 1

f(6) = 6


●

●

●

●

3.3 – Graphing Linear Functions

Identifying InterceptsThe graph of y = 4x – 8 is shown below.

The intercepts are: (2, 0) and (0, –8).

The graph crosses the y-axis at the point (0, –8).

Likewise, the graph crosses the x-axis at (2, 0).

This point is called the y-intercept.

This point is called the x-intercept.


Finding the x and y interceptsTo find the x-intercept:

let y = 0 or f(x) = 0 and solve for x.

To find the y-intercept: let x = 0 and solve for y

Given: 4 = x – 3y, find the intercepts

x-intercept, let y = 0

4 = x – 3(0)

4 = x

x-intercept: (4,0).

y-intercept, let x = 0

4 = 0 – 3y

4 = –3y

y-intercept: (0,-4/3).

–4/3 = y


Plotting the x and y intercepts

x

y

The graph of 4 = x – 3y is the line drawn through these points.

Plot the two intercepts

and (4, 0) and . 0, 4

3

(4, 0)

(0, )3

4

3.3 – Graphing Linear FunctionsVertical Lines

Graph the linear equation x = 3.

x y

3 0

3 1

3 4

Standard form as x + 0y = 3.

No matter what value y is assigned, x is always 3.

3.3 – Graphing Linear FunctionsHorizontal Lines

Graph the linear equation y = 3.

Standard form as 0x + y = 3.

No matter what value x is assigned, y is always 3.

x y

0 3

1 3

5 3

3.4 – The Slope of a LineSlope of a Line

2 1

2 1

2 1

change in

change in

y yym

xr

rise

x

un x x

x

The slope m of the line containing the points (x1, y1) and (x2, y2) is given by:

3.4 – The Slope of a LineFind the slope of the line through (0, 3 ) and

(2, 5). Graph the line.

5 3 21

2 0 2m

3.4 – The Slope of a LineFind the slope of the line containing the points (4, –3 )

and (2, 2). Graph the line.

2 ( 3) 5

2 4 2m

3.4 – The Slope of a Line

Only a linear equation in two variables can be written in slope-intercept form,

y = mx + b.

Slope-Intercept Form

M is the slope of the line and b (0, b) is the y-intercept of the line.

slope y-intercept is (0, b)

Find the slope and the y-intercept of the line

Example

Solve the equation for y.

3 2 11.x y


The slope of the line is 3/2.

The y-intercept is (0, 11/2).

Find the slope of the line x = 6.

Vertical line

Use two points (6, 0) and (6, 3).

Example

2 1

2 1

3 0 3

6 06

y ym

x x

The slope is undefined.


Find the slope of the line y = 3.

Horizontal line

Use two points (0, 3) and (3, 3).

Example

2 1

2 1

3 03

3 0 3

y ym

x x

The slope is zero.


Slopes of Vertical and Horizontal Lines

The slope of any vertical line is undefined.


The slope of any horizontal line is 0.

Appearance of Lines with Given Slopes

Positive SlopeLine goes up to the right

x

y

Lines with positive slopes go upward as x increases.

Negative SlopeLine goes downward to the right

x

y

Lines with negative slopes go downward as x increases.

m > 0m < 0


Appearance of Lines with Given Slopes

Zero Slope horizontal line

x

y

Undefined Slopevertical line

x

y


Parallel Lines & Perpendicular Lines

x

y


Two non-vertical lines are perpendicular if the product of their slopes is –1.

x

y

Two non-vertical lines are parallel if they have the same slope and different y-intercepts.



Are the following lines parallel, perpendicular, or neither?Example

The lines are perpendicular.



x + 5y = 5 –5x + y = –6

y = mx + b has a slope of m and has a y-interceptof (0, b).

Slope-Intercept Form

3.5 – Equations of Lines

This form is useful for graphing, as the slope and the y-intercept are readily visible.

Example

Graph

The slope is 1/4.

The y-intercept is (0, –3)

Plot the y-intercept.

Slope = rise over run.

Rise 1 unit; run 4 units right

The graph runs through the two points.

13

4y x


Example

Write an equation of the line with y-intercept (0, –5) and slope of 2/3.

y bmx

5)2

3(y x

25

3y x


The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line.

Point-Slope Form

)( 11 xxmyy

m is the slope


(x1, y1) is a point on the line

Find an equation of a line with slope – 2, containing the point (–11, –12). Write the equation in point-slope form and slope-intercept form.

Example

Point-Slope Form


)( 11 xxmyy y – (–12) = – 2(x – (–11))

y + 12 = –2x – 22 y = 2x – 34

y + 12 = – 2(x + 11)

Slope-intercept Form

Point-slope Form

10

1

)4(6

01

12

12

xx

yym

Example

1 1( )y xmy x

Point-Slope Form


Find an equation of the line through (–4, 0) and (6, –1). Write the equation using function notation.

1 1( )y xmy x

Example

Point-Slope Form


Example

Point-Slope Form


Write the equation of the line graphed. Write the equation in standard form.

2 1

2 1

5 3 2 1

2 ( 4) 6 3

y ym

x x

Continued.

●

●Identify two points on the line.

(2, 5) and(4, 3)

Find the slope.

Example3.5 – Equations of Lines

1 1( )y xmy x (2, 5) and(4, 3)


1 1( )y xmy x (2, 5) and(4, 3)

LCD: 3


Find an equation of the horizontal line containing the point (2, 3).

● The equation is y = 3.


Find an equation of the line containing the point (6, -2) with undefined slope.

●

What type of line has undefined slope?

The equation is x = 6.

Horizontal or Vertical

Documents

2.7 – Absolute Value Inequalities To solve an absolute value inequality, treat it as an equation. Create two inequalities, where the expression in the