3.1 Graphing ReviewSo far, we have Rearranged and Solved Equations.

Now we are going to Draw equations.

• An equation has an equals sign:

2x + 1 = 11

• We can make a graph if an equation looks like this:

y = 2x + 1✓ there are two letters✓ one letter is totally by itself

One way to Graph an Equation is to make a table

x y = 2x + 1 point

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

1 y = 2( 1 ) + 1 = 4 ( 1 , 5 )

2 y = 2( _ ) + 1 = _ ( _ , _ )

3 y = 2( _ ) + 1 = _ ( _ , _ )

allows equation to be solved

➀ get x-values

➁ plug-in for x, get value for y

➂ make points (x,y)

➃ plot points on grid

Determining Co-ordinates

• A ( __ , __ )

• B ( __ , __ )

• C ( __ , __ )

• D ( __ , __ )

• E ( __ , __ )

• F ( __ , __ )

Investigating Points A point (x,y) satisfies an equation if

The equation makes that point when the x-value is plugged in

Does (1,7) Satisfy y = 2x + 5 ? Does (-3,2) Satisfy y = -x + 5 ?

x = __ y = 2(__) + 5

when x = __ , y = __ → (__ , __)

x = __ y = -(__) + 5

when x = __ , y = __ → (__ , __)

∴ the point _____________ satisfy the equation

(it is/is not one of the points the eq’n creates)

∴ the point _____________ satisfy the equation

Investigating TablesDetermine if the tables will give the graph of a line.Hint: You will get a line if the y-values go up/down by the same amount.

x y x y

2 1 -3 1

4 5 0 -8

6 9 3 1

8 13 6 28

10 17 9 73

12 21 12 136

The following tables give the graph of a line.Fill in the y-values for these tables:

x y x y

2 5 -3 4

4 7 0 1

6 3

8 6

10 9

12 12

Using Variables that are not x and yA long distance plan charges $0.10 per minute when phoning New Zealand. The formula that models the cost of the call is C = 0.10m, where C is the total cost of the call, and m is the total number of minutes.

Develop a table of values for each minute up to 10 minutes of phone calls.

C = 0.10m

m C = 0.10mC = 0.10m Points

2 0.10(___) = 0.2 ( __ , __ )

4 0.10(___) = 0.4 ( __ , __ )

6 0.10(___) = 0.6 ( __ , __ )

8 0.10(___) = 0.8 ( __ , __ )

10 0.10(___) = 1 ( __ , __ )

The Cost of Calling New Zealand

One variable is alone

One variable is inside an expression

3.2 The Equation of a LineThere are many types of equations, and each type of equation creates it’s own unique graph.

For Example:

y = ⎮x⎮ y = x1/3 y = x2

Graphing Lines

y = mx + b... is the equation that always gives the graph of a line

Some Examples ...

note: In y = mx + b, m and b are numbers. The line changes as these numbers change.

Investigating m and b in y = mx + b Part 1• In y = mx + b, m and b are numbers. • The line changes as these numbers change.

Increasing my = x + 1 m = ___ b = ___ y = 2x + 1 m = ___ b = ___

y = 3x + 1 m = ___ b = ___ y = 8x + 1 m = ___ b = ___

What happens to the line as m gets bigger?

Investigating m and b in y = mx + b Part 2Decreasing my = x + 1 m = ___ b = ___ y = 0.5x + 1 m = ___ b = ___

y = 0.3x + 1 m = ___ b = ___ y = 0.25x + 1 m = ___ b = ___

What happens to the line as m gets smaller?

Investigating m and b in y = mx + b Part 3Negative my = x + 1 m = ___ b = ___ y = -x + 1 m = ___ b = ___

y = 0.3x + 1 m = ___ b = ___ y = -0.3x + 1 m = ___ b = ___

What happens to the line as m becomes negative ?

Based on your observations, what does changing m do to the graph?

Investigating m and b in y = mx + b Part 4Changing b

y = x + 1 m = ___ b = ___

y = x + 2 m = ___ b = ___

y = x + 0.5 m = ___ b = ___

y = x - 1 m = ___ b = ___

What happens to the lines as b changes?

Summaryfor y = mx + b ...

m is known as the ________, or _______ ___ ________, of the line.

If m is positive, the line is _______ on the left, and _______ on the right.

If m is negative, the line is _______ on the left, and _______ on the right.

As m becomes larger, the line becomes _______________.

b is know as the _____________, or _________ ________, of the line.

The _____________ is the point where the line crosses the __________.

Examples➀ Identify the slope and y - intercept

a) y = 3x + 2 b) y = -x c) y = 3 - 0.5x d) y = 13x −

15

m = ___ b = ___ m = ___ b = ___ m = ___ b = ___ m = ___ b = ___

➁ Write the equation a) m = 5, b = -1 b) m = − 1

2, b = 1 c) m = -4, b = 0

➂ Arrange the lines in order of steepness (the most steep goes first)a) m1 = 4, m2 = 7, m3 = 2, m4 = 2.5 c) m1 = -1, m2 = 2, m3 = -4, m4 = 3

b) m1 = -6, m2 = -3, m3 = -9, m4 = -2 d) m1 = -1/2, m2 = 1/7, m3 = 1, m4 = 1.5

3.3 The Rate of Change & SlopeProblem :

How can we determine the equation of the line from a graph?

✓ The equation of a line has two numbers to fill in:

y = mx + b

✓ Once you figure out m and b, just plug them into y = mx + b.

✓ The value for b is simply where the graph crosses the _____________ .

✓ Slope can be determined two ways:1) using rise/run 2)using two points on the line

Finding slope using rise/run

✓put dots at the corners the line goes through

• dots should be evenly spaced✓pick any two dots✓count, from dot to dot,

• how many we rise✓rise up, positive number✓rise down, negative number

• how many we run✓run right, positive number✓run left, negative number

★ SLOPE IS ALWAYS A FRACTION

m = b = y = ___x + ____

slope y - intercept

Finding slope using rise/run continued

m = b = y = ___x + ____ m = b = y = ___x + ____

Finding slope using two points on the line✓ If you know the coordinates of any two points on the line, and the y-

intercept, you can determine the equation of the line

Point A = (x1, y1) Point B = (x2, y2) y-intercept = b slope = (y2) - (y1) (x2) - (x1)

A(1,2) B(2,6) b = -2

A(-1,4) B(2,-5) b = 1

A(-4,5)

B(2,2)

b = 3

A(-4,-2) B(-8,-3)

b = -1

3.4 Graph using y = mx + bHow can we quickly graph a line without creating a table of values?

• all you need to graph a line is the equation y= mx + b

• m is the ________________.

• b is is where the line ____________________________.

• b is called the _________________.

Graphing a line using the slope and y-intercept:Steps: - Rewrite y = mx + b so that m is a fraction with negative signs on top.- Plot the y-intercept.- From the y-intercept Count up/down for rise , Count right for run- Draw another point where you finished counting.- Connect the points with a line.

y = 2x + 1 y = -4x + 2

y = −12x −1 y =

13x +1

3.5 Graph using Ax + By + C = 0• Lets rearrange y = mx + b so that everything is on one side:

y = mx + b ----> Ax + By + C = 0

• Ax + By + C = 0 will still give you the graph of a line• We can graph quickly by using Ax + By + C = 0 and intercepts

What are Intercepts ?• An Intercept is a point where a graph crosses an axis.• All types of graphs can have Intercepts.

x - intercept y - intercept

- crosses __________ - crosses __________

- coordinates ( __ , __ ) - coordinates ( __ , __ )

• A Linear Graph always has one x-intercept, and one y-intercept.

Identify the Intercepts:

x-int : ____ y-int : ____ x-int : ____ y-int : ____

x-int : ____ y-int : ____ x-int : ____ y-int : ____

slope-intercept form standard form

Graphing a line using intercepts:• If you want to graph from the equation Ax + By + C = 0, you will need to use intercepts.

Step 1 - Find the x-interceptFor an x-intercept, the y value is always ___. ---> ( __ , __ )So plugging in a value of ___ for y, and solving for x, gives the x-intercept.

Step 2 - Find the y-interceptFor an y-intercept, the x value is always ___. ---> ( __ , __ )So plugging in a value of ___ for x, and solving for y, gives the y-intercept.

Step 3 - Connect the points

Draw the graphs for the stated equations :5x + 2y - 10 = 0x - int y-int

3x - 4y - 12 = 0x - int y-int