Chapter 8

KRUSKA

Solve each equation. 1. −20 = −4x − 6x 2. 6 = 1 − 2n + 5 3. 8x − 2 = −9 + 7x 4. a + 5 = −5a + 5 5. 4m − 4 = 4m 6. p − 1 = 5p + 3p − 8 7. 5p − 14 = 8p + 4 8. p − 4 = −9 + p 9. −8 = −(x + 4) 10. 12 = −4(−6x − 3)

Answers:

1. x = 2

2. x = 0

3. x = -7

4. x = 0

5. x = NS

6. x = 1

7. x = -6

8. x = NS

9. x = 4

10. x = 0 Monday KRUSKA

Chapter 8 – Functions and Graphing:

Week 8 – April 8th

8-1 Relations and Functions

8-3 Graphing Linear Relations

QUIZ 8A

Week 7 – April 15th

8-4 Equations as Functions

8-6 Slope

8-7 Intercepts

QUIZ 8B

Week 6 – April 22nd

8-8 Systems of Equations

8-9 Graphing Inequalities

Chapter 8 Test

ON MONDAY, APRIL 29TH WE BEGIN PREPERATION

FOR YOUR FINAL EXAM!!!

Monday KRUSKA

SLOPE-INTERCEPT FORM:

y=mx+b

POINT-SLOPE FORM:

y – y1 = m(x – x1)

SLOPE FORM:

y2 – y1 = m x2 – x1

Monday KRUSKA

NONE - HAPPY EASTER!

Graphing points

Monday KRUSKA

Objective: To use table and graphs to

represent relations and functions

Warm-Up: 1. 3x - 2 = 13 2. 6 - 5x = 21

3. (2x + 7) = x + 3

3

4. -12x > -24

5. 7x < 2x - 25

Answers: 1. x = 5

2. x = -3

3. x = -2

4. x < 2

5. x < -5

Tuesday KRUSKA

NOTES: Domain (x) - Set of FIRST (1st) coordinates Range (y) - Set of SECOND (2nd) coordinates

Relation (x,y) - Set of ordered pairs or

coordinates

Function – A relation in which each number of the domain is paired with exactly one element in the range No duplicates!

Tuesday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Find a “RULE” (or an equation with an “X” and a “Y”)

Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Insert the “RULE” (or an equation with an “X” and a “Y”)

day KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Insert the “X” values

Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Insert the “X” and “Y” values

Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Out comes the “Y” value. Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Now, combine them and you get a

coordinate (X,Y) for a point on the line Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Insert another “X” value (Do it again for the second coordinate point)

Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Out comes the coordinates (X,Y) for

the second point on the line ) Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Insert a third “X” value (Do it again for the third coordinate point)

Friday KRUSKA

NOTES: Do you

remember

the…

Function

Machine?

Out comes the coordinates (X,Y) for

a third point on the line Friday KRUSKA

NOTES Do you

remember

the…

Function

Machine?

If the function machine works,

then the three coordinate points

will line up to form a straight line! Friday KRUSKA

IT WORKED!!

Friday KRUSKA

NOTES:

Tuesday KRUSKA

NOTES: What are the points on this graph?

From top to bottom…

(2, 5)

X = 2 and y = 5

(0, -1)

X = 0 and y = -1

(-1,-4)

x= -1 and y = -4

Tuesday KRUSKA

NOTES: What are the Quadrants??

Tuesday

“I C Quads”

KRUSKA

P374-375 (6–28 EVEN)

Graphing points and naming

quadrants

Points on the coordinate plane

Tuesday KRUSKA

Objective: To find solutions for relations with two

variables and graph the solutions

Warm-Up: Name the Range & Domain of:

1. (5,6)

2. (-4,3)

3. (0.3,2.1)

4. (13,-49)

5. (5, 7½)

Answers: 1. D=5 R=6

2. D=-4 R=3

3. D=0.3 R=2.1

4. D=13 R=-49

5. D=5 R=7 ½

BLOCK KRUSKA

Section A – Individual

▪ P 375 (17 - 27 ODD)

▪ P 376 (29-39 ODD)

Section B – With Mr. Kruska

▪ P388 - 389 (9-36 x3)

▪ WS - Graphing with Charts

Section C – KHAN

▪ Graphing Linear Equations

BLOCK KRUSKA

NOTES: Graph the following equation using a

function chart: y = −x + 2

X Domain

Y = −x + 2 Rule

Y Range

(X,Y) Relation

-1 Y = - (-1) +

2

3 (-1, 3)

0 Y = - (0) + 2 2 (0, 2)

1 Y = - (1) + 2 1 (1, 1)

BLOCK KRUSKA

NOTES: Graph by making a table!

BLOCK KRUSKA

BLOCK KRUSKA

Section A – Individual

▪ P 375 (17 - 27 ODD)

▪ P 376 (29-39 ODD)

Section B – With Mr. Kruska

▪ P388 - 389 (9-36 x3)

▪ WS - Graphing with Charts

Section C – KHAN

▪ Graphing Linear Equations

BLOCK KRUSKA

Objective: To use table and graphs to

represent relations and functions

Warm-Up: 1. 3x - 2 = 13

2. 6 - 5x = 21

3. (2x + 7) = x + 3

3

4. -12x > -24

5. 7x < 2x - 25

Answers: 1. x = 5

2. x = -3

3. x = -2

4. x < 2

5. x < -5

Friday KRUSKA

NOTES:

What is the formula for finding the

SLOPE of a line? (Slope Form)

What is the formula for finding the

Y-Intercept of a line? (Slope-Intercept

Form)

KRUSKA

y2 – y1 = m x2 – x1

y=mx+b Friday

NOTES:

What is the formula for finding the

SLOPE of a line? (Slope Form)

(2, 3) & ( 4, 1)

y2 – y1 1 – 3 = -2 = -1 = m

x2 – x1 4 – 2 2 KRUSKA

y2 – y1 = m x2 – x1

Friday

NOTES:

What is the formula for finding the

Y-Intercept of a line? (Slope-Intercept

Form)

KRUSKA

y=mx+b

y = 3x + 2

Slope 3/1

Y-Intercept (0,2)

y = -1/3x - 1

Slope -1/3

Y-Intercept (0,-1)

Friday

NOTES:

What is the formula for finding the X &

Y Intercepts of a line? (Standard Form)

KRUSKA

Ax + By = C

2x + 3y = 6

X-Int

• 2x + 3(0) = 6

• 2x = 6

• x = 3

• (3,0)

2x + 3y = 6

Y-Int

• 2(0)+ 3y = 6

• 3y = 6

• y = 2

• (0,2) Friday

NOTES: Standard to Slope Intercept

Friday KRUSKA

NOTES: Function Notation - f(x):

The symbol f(x) is read “f of x” Any letter can be replaced

with the “f” and it is still a Function Notation

g(x) is read “g of x” is… Replace the (x) value to

find the “y” What is g(x) = 3x + 4 if

g(3)? g(3) or “y” = 13

Friday KRUSKA

p394 (4-30 EVEN)

Solving for Y-Intercept

Slope Intercept Form

Friday KRUSKA

Objective:

To find the slope of a line

Warm-Up: P 399 (1, 2, 7 , 8, & 9)

Tuesday KRUSKA

Do-It-Yourself Slope?!

Tuesday KRUSKA

NOTES:

SLOPE FORM:

The letter “m” is used to show the

slope of a line

y2 – y1 = m x2 – x1

I am

”SLOPE MAN”!

Tuesday KRUSKA

4 12

6 14

9 17

11 19

What is the SLOPE?

Now GRAPH the

relation (or chart) by

using a function of a

linear line

Tuesday

y2 – y1 = m x2 – x1

KRUSKA

Slope Form:

y2 – y1 = m x2 – x1

1. (2, 4) (3, 1)

2. (4, 2) (0, 3)

3. (-3, -1) (-4, 2)

Tuesday KRUSKA

P 402-403 (6-20 EVEN)

Point Slope Form

Tuesday KRUSKA

Objective:

To graph a linear equation using

the x-intercept and y-intercepts or using the slope and y-intercept

Warm-Up:

Find the slope:

1. (-4, -2) & (5, 3)

2. (1, 4) & (5, -2)

3. (0,7) & (2, -7)

4. (0, 0) & (4, -4)

5. (-4, -4) & (-5, -3)

Answers: 1. 5/9

2. -3/2

3. -7

4. -1

5. -1

Block KRUSKA

Block KRUSKA

Block

NOTES:

KRUSKA

What is the slope and y-intercept of the

graph of y = 2x 3?

◦ Slope (m) is -2/1

◦ Y-Int. (b) is -3

Block KRUSKA

Convert the following equation from slope intercept form

to standard form.

In other words, if the equation is rewritten to look like Ax+By=C, what

are the values of A, B, and C?

y=−x−2

A = ?

B = ?

C = ?

Answers:

A = 1

B = 1

C = -2

Block KRUSKA

P 409 (9 - 33 x3)

Converting between

slope-intercept and

standard form

Block KRUSKA

Objective:

To solve systems of linear

equations by graphing

Warm-Up:

Graph using the X-intercept

and Y-intercept:

1. y = 6 - x

2. y = x - ½

Answers: 1. (0,6) & (6,0)

2. (0, - ½) & (½, 0)

Monday KRUSKA

Notes:

If you can graph a straight line, you can solve

systems of equations graphically!

The process is very easy. Simply graph the

two lines and look for the point where they

intersect (cross).

Monday KRUSKA

Notes:

Line A: 2x + 2y = 6 Line B: 4x - 6y = 12

Solution - (3, 0)

Monday KRUSKA

Notes:

Check: Since the two lines cross at (3,0), the solution

is x = 3 and y = 0. Checking these value shows that this answer is

correct. Plug these values into the ORIGINAL equations

and get a true result.

Monday

4x - 6y = 12

4(3) - 6(0) = 12

12 - 0 = 12

12 = 12 (check)

2x + 2y = 6

2(3) + 2(0) = 6

6 + 0 = 6 6 = 6 (check)

KRUSKA

Notes:

Whenever two equations have

the same slope they will

be parallel lines.

Parallel lines NEVER intersect.

Therefore, the system of

equations will NOT have a

solution!

Monday

HINT!

KRUSKA

Notes:

Line A: y = - ½ x - 6 Line B: y = - ½ x +4

Solution -

No Solution

because they

never touch…

they are parallel

Monday KRUSKA

Notes: Let’s Review!

Monday KRUSKA

Notes: Let’s Review!

Monday KRUSKA

Notes: Let’s Review!

Monday KRUSKA

Notes: Let’s Review!

Monday KRUSKA

P 415 (6 - 20 EVEN)

Graphs of inequalities

Monday KRUSKA

Objective:

To graph linear inequalities

Warm-Up:

Graph these linear equations:

1. Y = x + 1 & y = -x + 3

2. 2x - y = 6 & x = y + 2

Answers: 1. (1, 2)

2. (4, 2)

Tuesday KRUSKA

Notes:

Line A: 2x + 2y < 6 Line B: 4x - 6y > 12

Solution - Shade!

Monday KRUSKA

Notes:

Line A: y < - ½ x - 6 Line B: y > - ½ x +4

Solution - Shade!

NO SOLUTION?

Monday KRUSKA

Notes:

Tuesday KRUSKA

P 421 (6 - 26 EVEN)

Graphing linear

inequalities

Tuesday KRUSKA

What is an equation in slope-intercept form for the line that passes through the points (1, 3) and (3, 1)?

Substitute the two given points into the slope form to find the slope of the line.

Then substitute the slope and the coordinates of one of the points into the slope-intercept form to find b.

Use slope-intercept form. 3 = 2(1) + b Substitute 2 for m, 1 for x, and 3 for y. 5 = b Solve for b. Substitute the slope and y-intercept into the slope-

intercept form. y = mx + b Use slope-intercept form. y = 2x (5) Substitute 2 for m and 5 for b.

KRUSKA

Slope-Intercept Form y = mx + b

1. (2, 4) (3, 1)

2. (4, 2) (0, 3)

3. (-3, -1) (-4, 2)

KRUSKA

Section A - Individual p 781 (2 - 32 EVEN)

Section B - Kruska p 424 (9-13 ALL)

p 425 (20-46 EVEN)

p 426 (48-58 EVEN)

Section C - Groups Climb the Cliff

Block KRUSKA