Last time fill in some definitions of equation stuff from 1-3



[pic]

The Coordinate Plane

Section 4.1

Warm up

Use the formula F=[pic]C + 32 to find the following:

1. What Celsius temperature is equivalent to 32 degrees Fahrenheit?

2. What Fahrenheit temperature is equivalent to 225 degrees Celsius?

3. What Celsius temperature is equivalent to -40 degrees Fahrenheit?

Write down three or more things that you remember about graphing.

(the picture may help you remember)

Coordinate Plane:

Origin:

X-axis:

Y-axis:

Ordered Pair:

X-coordinate:

Y-coordinate:

Example: Write the ordered pairs that correspond to points A, B, and C.

A.

B.

C.

Example: Plot the coordinate points in the coordinate plane.

A= (3,4) D = (-1,-3)

B = (-2,5) E = (-2,0)

C = (3,5) F = (-2,-3)

Quadrants for the Coordinate Plane:

Signs of the coordinates

in each quadrant:

I.

II.

III.

IV.

Example: Name the quadrant the point is in the coordinate plane.

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

3. (-1,-3) 4. (-3,6)

You try:

Scatter Plot:

Example: The amount (in millions of dollars) spent in the U&S on snowmobiles is shown in the table. Make a scatter plot and explain what it indicates.

Year |1990 |1991 |1992 |1993 |1994 |1995 |1996 | |Spending |322 |362 |391 |515 |715 |924 |970 | |

Calculator:

To be completed on the calculator:

Points to Remember:

Homework:

Graphing Linear Equations with a Chart

Warm up A

B

1. Write the ordered pair for each point. C

D

E

2. Find the value of y when x = -3.

a. y = x – 7 b. y = -5x + 1

3. Make an input/ output table . Use the domain (values for x) { -1, 0, 1, 2}

Y = 3x - 2

Function form:

Rewrite each equation by solving for y. MEANS PUT IN FUNCTION FORM

1. 2x + y = 10 2. 6x – 3y = -3

3. -3x + y = 12 4. 2x + 3y = 6

5. 5y – 2x = 15 6. 4x – y = 18

7. 2x – y = 7 8. 6x + 3y = 18

9. 4y – 3x = -28

Your Turn:

Rewrite each equation in functional form (isolate the y).

11) –x + y = 6 12) x + y = –2

13) x – y = –2 14) –2x + y = –4

15) 4x + 2y = 10 16) –9x + 3y = –6

17) –2x – 4y = 12 18) 5x + [pic]y = 0

19) –2x + ½ y = 7 20) 5x – 3y = 21

Solution of an equation:

Example: Determine whether the ordered pair is a solution of x + 2y = 5.

STATE YES OR NO AND PROVE IT BY SHOWING ALL WORK

1. (1,2) 2. (7,-3)

Give it a try: Determine whether the ordered pair is a solution of 2x + y = 1.

STATE YES OR NO AND PROVE IT BY SHOWING ALL WORK

1. (-3,7) 2. (3,-7) 3. (½,0)

Example:

Find three ordered pairs that are solutions of -2x + y = -3.

Steps:

1. Rewrite the equation in function

form to make it easier to substitute values.

2. Choose any value for x and

substitute it into the equation to find y.

3. Select a few more values of x and repeat step 2.

Example: Use a table of values to graph y = 3x – 2.

Steps:

1. Make a table of values

2. Plot the points

3. Connect the points

Use a table of values to graph 4y – 2x = 8.

Summary: How to Graph a Linear Equation using a Table of Values:

Step 1:

Step 2:

Step 3:

Use number 27 on the problems below and graph

it here

Graphing Horizontal and Vertical Lines

Warm up

Find the value of y when x = 6 (Means plug in 6 for x and tell me what y is)

1. y = ½ x – 7 2. y = -3x + 1

Circle the correct answer for each question

Evaluate: 3 + 4 [pic] 52 + 7

a) 182 b) 110 c) 39 d) 224

Evaluate [pic]

a) -2 b) 2 c) -[pic] d) [pic]

Evaluate [pic] + ([pic])

a) [pic] b) -[pic] c) [pic] d) [pic]

Find the product: [pic](9)[pic]

a) -[pic] b) [pic] c) -[pic] d) -[pic]

What is the property [pic]

a) Commutative Property of Multiplication

b) Associative Property of Multiplication

c) Identity Property of Multiplication

d) Inverse Property of Multiplication

Horizontal Lines:

• Direction:

• Equation:

• Every ____ coordinate is the ______

*Note* The graph of the line crosses

Example: Graph the line y = -5.

X Y

Vertical Lines:

• Direction:

• Equation:

• Every ______ coordinate is the ________

*Note* The graph of the line crosses

Example: Graph the line x = -5

X Y

A trick to remember:

Horizontal Vertical

Graph the following equations:

x = -1 y = -

[pic] [pic]

x = 6 y = 4.5

[pic] [pic]

Determine whether the given ordered pair is a solution of the equation.

1) x = -4 (4, -4) 2) y = 5 (-3, 5) STEPS TO DECIDE

Find three ordered pairs that are solutions of the equation.

Points to remember:

Homework:

Graphing Lines Using Intercepts

Warm up

Rewrite each equation in function form.

1. 2x - 4y = 24 2. 21x + 3y = -9

3. Find the value of y when x = 0 in the equation y = ½ x – 7. (plug in zero for x and figure out what why is)

4. Find the value of x when y = 0 in the equation y = -3x + 1. (plug in zero for y and figure out what x is)

5. On the graph at the right give the coordinates of points

A and B. A

B

6. On the graph at the right give the coordinates of points

A and B

A

B

NOTES:

x-intercept:

y-intercept:

Function Form:

Standard Form:

Steps for using an equation to find the x and y intercepts:

x-intercept: y-intercept:

Find the x-intercept and y intercept of the equation 2x + 3y = 6.

x-intercept: y-intercept:

Find the x and y-intercept of the equation 3x – 4y = 12.

x-intercept: y-intercept:

Your Turn:

Find the x and y-intercept of the equation 4x – 8y = 24

x-intercept: y-intercept:

Steps to making a quick graph: Not using a table of values but using intercepts

1.

2.

3.

4.

Example: Graph the equation 3x + 2y = 12.

x-intercept: y-intercept:

Example: Graph the equation [pic]

x-intercept: y-intercept:

Give it a try: Graph the equation x – 6y = 6.

x-intercept: y-intercept:

Points to remember:

Class Work/Homework: This worksheet.

Graphing using the x– and y– intercepts.

1. x-intercept = (5,0) 2. x-intercept =( –4,0)

y-intercept =(0 –2) y-intercept = (0,–1)

3. 6x + 5y = –30 4. 3y – 2x = 12

5. 4x – 7y = 14 6. 2x + y = 8

7. 3x – y = –6 8. [pic]

9. y = 6 – x 10. [pic]

11. y – 2x = 5 12. [pic]

Slope of a Line

Warm up

Identify the x and y coordinate of each ordered pair.

1. (-3,4) 2. (0,-7) 3. (5, 0) 4. (-9,-2)

Slope:

Formula for Slope: [pic]

Example: Find the slope of a hill that has a vertical rise of 40 ft and a horizontal run of 200 ft. Let m represent slope. Draw a picture.

When finding the slope of a line that passes through two specific points (x[pic],y[pic]) and (x[pic],y[pic]), the vertical run and horizontal run are found by:

[pic]

Example: Find the slope of the line that passes through the points (1,0) and (3,4).

[pic] [pic]

Example: Find the slope of the line that passes through the points (3,5) and (1,4).

[pic] [pic]

Give it a try:

1. Find the slope of the line that passes through the points (2,6) and (4,3).

2. Find the slope of the line that passes through the points (2,7) and (1,3).

We have only worked so far with slopes that are positive. But, slopes can also be negative, zero, or not even exist!!!

Positive Slope Negative Slope Zero Slope No Slope

[pic][pic][pic][pic]

Example: Find the slope of the line that passes through the points (0,3) and (6,1).

Example: Find the slope of the line that passes through the points (2,4) and (-1,5).

Give it a try: Find the slope of the line that passes through the points (0,9) and (4,7).

Example: Find the slope of the line that passes through the points (1,2) and (5,2).

What will this line look like?

Example: Find the slope of the line that passes through the points (5,-1) and (5,3).

What will this line look like?

Direct Variation

Section 4.6

“In this lesson, we you will … write and graph equations that represent direct variation.”

Warm up

Find the slope that passes through the following points:

(3,-4) (6,-9) (5, 9) (5,12) (3,12) (5,12)

If y = mx solve for m when x = 3 and y = 18. Then, use the answer for m along with x = 20 to solve for y.

Direct variation process:

1) Write the basic formula ________________________.

2) Use what you are given to find __________________.

3) Write the new formula with _________ and variable _______ and ____.

4) Find the question asked.

Example: The variables x and y vary directly x=7 when y = 21. Write the equation that relates x and y. Evaluate when x = 4.

Example The variables x and y vary directly x=6 when y = 30. Write the equation that relates x and y. Evaluate when x = 2.

Example: The variables x and y vary directly x=42 when y = 2. Write the equation that relates x and y. Evaluate when x = 3.

Word problem fun:

1. A certain car uses 15 gal of gasoline in 3h. If the rate of gasoline consumption is constant, how much gasoline will the car use on a 35-hour trip?

2. The distance traveled by a truck at a constant speed varies directly with the length of time it travels. If the truck travels 168 mi in 4 h, how far will it travel in 7h.

Example: The variables x and y vary directly. One pair of values is x = 5 and y = 20. Write and equation that relates x and y and then find the value of y when x = 12.

Example: The variables x and y vary directly. One pair of values is x = 2 and y = 6. Write and equation that relates x and y.

Write the general equation relating x and y

Give it a try: Use the values x = 3 and y = 21 to write a direct variation model that relates x and y.

Points to Remember:

[pic]

Graphing Lines Using Slope-Intercept Form

Warm up

1. Rewrite in function form:

a. 12x + 4y = 24 b. 7y – 14x = 28

c. 3x – 6y = –18 d. 3x + 4y = 16

2. Graph the equation using 3. Graph the equation using an

intercepts. 2x + 3y = 6 x|y chart (input output table) y = 2x + 3

Function form:

Slope-intercept form:

y = _______ x + _______

Slope: Y intercept:

The fun part is we no longer need two points to find the slope or need to plug in x=0 to find our y-intercept. Now, we can just look at the slope-intercept form!!

Examples: Find the slope and y-intercept

y = 2x – 3 y= ½ x+2 y = -3x + 5

m = m = m =

b = b = b =

y = x + 6 y = -2x – 5 2x + 4y = 12

m = m = m =

b = b = b =

How to graph a line in slope-intercept form:

Step 1: Put equation into _____________________

Step 2: Identify the ___________________ as a _________________

Step 3: Identify the ___________________________

Step 4: Graph the ________________________

Step 5: Graph the ______________ as ____________ over _______________

Step 6: Repeat step 5.

Ex: Graph the equation y= ½ x-3. Ex: Graph the equation y = 2x + 1

m = m =

b = b =

Parallel lines:

Ex: Which of the following are parallel lines?

a) –x+2y=6 b) -x+2y=-2 c) x+2y=4

Ex: Which of the following are parallel lines?

a) 3x+2y=6 b) 3x-2y=6 c) 6x+4y=6

Functions and Relations

Section 4.8

“In this lesson, you will … graph a linear equation using slope-intercept form.”

Warm up

A) Identify the real-life meaning of slope in each of the following graphs:

B) A store sells video cassettes for $5 each. The cost c of buying n video cassettes can be modeled by the equation c=5n. What values can n be? What values can c be? (Hint: Can we have negative cost? Can we have zero? Can we have negative video cassettes? Can we have zero?)

B) Identify the slope and y-intercept of the following:

1) 4x – y = 5 2) y = 3x -7

Recall: What is a function? What was the input? What was the output?

A rule for functions: A group of ordered pairs is a function if for every ____________ there is exactly one ______________.

Ex: Decide whether the following show functions.

1 2 1 5

2 4 2 7

3 5 3 9

4 4

Think:

Because each input “x” has exactly one output “y,” we can use a __________________

________________ test for functions. That is, if a vertical linear intersects the graph at more than one point, it ______________ a function.

Hint: Use your pencil!!!!!

Ex:

Ex: Identify the following as a function or not.

[pic]

When we are working with functions, we use what’s called “function notation.” All that this means is that we use ________ in place of our normal “y.”

xy-notation: y = 3x +2 function notation: f(x) = 3x +2

Ex: Evaluate f(x) = 2x – 3 when x = -2.

Ex: Evaluate f(x) = 4x + 5 when x = 2.

Ex: Evaluate g(x) = x[pic] when x = -3.

Every function that we have talked about and graphed in the past has been a straight line. When a function has the appearance of slope-intercept form, it is said to be _______________________________.

Ex: Graph f(x) = -½x + 3.

Ex: Graph f(x) = 4x - 3

Domain: Range:

Evaluate the function when x = 2, x = ½, x = -3 f(x) = 6x – 5

[pic]

-----------------------

[pic]

y = ½

( , )

( , )

( , )

x = -13

( , )

( , )

( , )

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