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Unit 2 – Solving Equations 1

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Unit 3

Unit 4

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Unit 10

– Unit 3a – [Graphing Linear Equations]


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Unit Calendar

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY

October 6 7 8 9 10

Graphing from a

Table

Slope

Slope

QUIZ

Slope Intercept

Form

Parent Function

And Transformations

13 14 15 16 17

Student Holiday Slope Intercept

Form

QUIZ

PSAT Standard Form and

Intercepts

Standard to Slope

Intercept Form

20 21 22 23 24

Solving for Slope

Intercept Form

Review

TEST

…

…

Essential Questions

What does it mean for relationship to be a linear function?

How can linear functions be used to model problem situations?

What are the critical features of a graph and how can they help me solve this problem?

How can a value of a function be represented?

What is the meaning of slope and how do I find it in this situation?


Critical Vocabulary

Linear Function

Parent Function

Slope

x-Intercept / Zero

y-intercept

Direct Variation

Parallel

Perpendicular


Graphing from a Table

Examples:

Equation: 𝑦 = 3𝑥 – 5 Table Graph

x y

-2

-1 0

1 2

3 . . .

. . .

Equation: 𝑦 = −2𝑥 + 6

Table Graph

x y

-2

-1 0

1

2 3 . . .

. . .

To babysit my younger brother, my mom gives me $5 and an extra $2 per hour.

Equation: 𝑦 =

Table Graph x y

-2 -1

0

1 2

3 . . .

. . .

Equation: 𝑦 = −2

3𝑥 + 5

Table Graph x y

-6 -3

0

3 6

9 . . .

. . .


Practice:

Equation: 𝑦 = 2𝑥 – 4

x y

-2

-1 0

1 2

3 . . .

. . .

Equation: 𝑦 = −4𝑥 + 3

x y

-2

-1

0 1

2 3 . . .

. . .

With no money in my bank, I started saving $3 every week.

Equation: 𝑦 =

x y

-2 -1

0

1 2

3 . . .

. . .

My box of chocolates had 5 pieces in it. I ate one piece each class period.

Equation: 𝑦 =

x y

-2 -1

0

1 2

3 . . .

. . .

Equation: 𝑦 =1

3𝑥– 4

x y

-6 -3

0 3

6 9 . . .

. . .

Equation: 𝑦 = −3

2𝑥 + 4

x y

-4

-2 0

2 4

6 . . .

. . .


Slope

Slope: The slope of a line is the measure of how ________ the line is. Slope is directly related to the ______ ___ _______ in a rule.

Slope Formula: m = 𝒚𝟐 − 𝒚𝟏

𝒙𝟐 − 𝒙𝟏

𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚

𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙

𝑹𝒊𝒔𝒆

𝑹𝒖𝒏

Positive Negative Zero (0) Undefined

Increases from Left to Right

Decreases from Left to Right

Stays the same (constant) from Left to Right

Not a function, so no definition for slope


Examples: Slope Formula: m = 𝒚𝟐 − 𝒚𝟏

𝒙𝟐 − 𝒙𝟏

Table 2 - Points Graph Situation

x y -2 12

-1 10

0 8 1 6

2 4

( -2 , 12 ) and ( 1 , 6 )

I have $8 to spend on snacks during the week. Each snack I purchase costs $2.

x y

-4 -1 -2 2

0 5 2 8

4 11

( 0 , 5 ) and (-2 , 2 )

Every 2 months I have to pay a $3 charge to keep my account open. Originally, I had to pay a $5 fee to open the account.

x y

-2 4 -1 4

0 4 1 4

2 4

( -1 , 4 ) and (2 , 4 )

A solar powered go cart began moving at 4 mph and never changed speed.

x y

-3 -2 -3 -1

-3 0

-3 1 -3 2

( -3 , -2 ) and ( -3 , 1 )

N/A This is not a function, so it does not have a situation to match it.


Practice:

x y

-4 -14

-2 -7 0 0

2 7 4 14

m =

( 1 , 3 ) and ( 7 , 6 )

m =

m =

I am paying my friend back $4 every week from the $20 I borrowed from her.

m =

m =

m =

x Y

1 -2 2 -2

3 -2 4 -2

5 -2

( 0 , 0 ) and ( 9 , -6 )

m =

m =

x y

-2 -3 -1 2

0 7 1 12

2 17

Benjamin got a $40 gift card to Starbucks for his birthday. His favorite drink costs $3 each time he goes. m =

( 4 , -1 ) and ( 4 , 9 ) m =

Every 6 months I pay my car insurance $131, plus I paid an intial fee of $25 when I signed up. m =

m =

x y -3 -4

-2 -3 -1 -2

0 -1

1 0

m =

m =

( 4 , -5 ) and ( -2 , -7 )

m =

m =

x y

-5 17

-3 11

-1 5

1 -1 3 -7


Slope Intercept Form

Slope-Intercept Form: The equation of a line written in a form that includes both the ________ and the ___ - _____________.

(m) (b)

Examples:

𝑦 = 3𝑥– 5

𝑦 = − 𝑥 + 7

𝑦 = 4𝑥

𝑦 =1

4𝑥 − 5

𝑦 = −3

2𝑥 + 7

𝑦 =2

5𝑥


Special Cases: Horizontal and Vertical Lines

H O Y

Horizontal Line

Zero Slope

0

5

𝑦 = constant

𝑦 = 2

V

Vertical Line

U

Undefined Slope

3

0

X

𝑥 = constant

𝑥 = −4

Examples:

𝑦 = −2

𝑥 = −2

𝑦 = −2𝑥


Practice:

𝑦 = 2𝑥 + 3

𝑦 =2

3𝑥 + 1

𝑦 = 6

𝑦 = −3𝑥

𝑦 = 𝑥– 5

𝑦 = −4

3𝑥– 1

𝑥 = 7

𝑦 = −3

2𝑥

𝑦 = −4𝑥 + 9


Linear Parent Function and Transformations

Transformations: The parts of an equation that change the shape or position of a graph/line.

Slope / Steepness: Ignoring the negative, if the slope is greater than one then the line is steeper.

𝑦 = 𝑥 𝑦 = 4𝑥 𝑦 =

5

2𝑥

**Improper Fractions Ignoring the negative, if the slope is less than one then the line is less steep.

𝑦 = 𝑥 𝑦 =

1

2𝑥

**Proper Fractions


Tran SL ation: If there is a constant (regular number with no variable), SL ide the change is a vertical shift or slide of the line.

𝑦 =

1

2𝑥 𝑦 =

1

2𝑥 + 3

Re FL ection: If the coefficient (number in front) of x is negative,

FL ip the change is a vertical flip of the line.

𝑦 =

1

2𝑥 + 3 𝑦 = −

1

2𝑥 + 3

Parent Function: The most basic form of a graph (No Transformations)

Linear Parent Function: 𝑦 = 𝑥 “𝑦 = +1𝑥 + 0”

No Reflection, Slope of 1, No Translation


Examples:

𝑦 = 4𝑥 − 3

Steepness: More Steep Less Steep No Change

Translation: Shift Up Shift Down No Change

Reflection: Reflection No Change

Parent Function

Transformed Equation

𝑦 = −

2

5𝑥 + 1



Reflection: Reflection No Change

Parent Function


Equation Steepness Translation Reflection

𝑦 =4

3𝑥 + 5

𝑦 = −3𝑥

𝑦 = −𝑥 − 27


Practice:

𝑦 =

4

3𝑥 + 1



Refelction: Reflection No Change

Parent Function


Equation Steepness Translation Reflection

𝑦 = −6𝑥 + 5

𝑦 =2

7𝑥 − 1

𝑦 = −𝑥

𝑦 = −3𝑥 − 8

𝑦 =3

2𝑥 − 6

𝑦 = 𝑥 + 4

𝑦 =1

4𝑥

𝑦 = 𝑥 − 35


Slope Intercept Form (Applications)

Examples: I owe my friend $10 and I plan to pay them back $2 each day. The equation that models this situation is

𝑦 = 2𝑥 – 10

A) What does the y represent?

B) What does the x represent?

C) What does the 2 represent?

D) What does the 10 represent?

E) Why is the 10 negative in this problem?

F) Would this be continuous or discrete?

G) Graph the equation

H) What is a reasonable domain for this problem?

Gina has a $30 gift card to Starbuck’s. Every time she buys a coffee she is charged $3.

A) Write an equation in Slope-Intercept Form to represent the problem

B) Why did you choose the sign of 3 to be positive or negative?

C) If we doubled the constant in the equation, what would that represent in the problem?

D) What is a reasonable domain for this situation?

E) Is this situation discrete or continuous?


Practice:

Chris does a lot of babysitting. When parents drop off their children and Chris can supervise at home, the

hourly rate is $3. If Chris has to travel to the child’s home, there is a fixed charge of $5 for transportation in

addition to the $3 hourly rate.

A) Write an equation for when he watches the

kids at home?

B) Write an equation for when he needs to travel to watch the kids?

C) Graph both Lines

D) Will a parent dropping off their kid ever pay more than if Chris has to travel to their home?

E) If in 3 years, Chris has a new equation, y = 5x + 10, what has changed in this problem?

The charges for Anderson’s Plumbing can be modeled by the following equation, where, C, is the total cost for

plumbing services that last, h, hours:

40 25C h

A) What does the 25 represent in this situation? B) What does the 40 represent in this situation?

The cost of dinner at a sweet 16 party is $300 plus $10 per person.

A) What is the equation that models this situation?

B) What does the y-intercept represent? C) What does the slope represent?

D) Find the cost for 50 guests.


Laura is on a 2 day hike in the Smoky Mountains. She hiked 8 miles the first day and is hiking at a rate of

3 mi/h on the second day. Her total distance is a function of the time she hikes.

A) What is the equation that models this situation?

B) What does the y-intercept represent? C) What does the slope represent?

D) What will be Laura’s total distance if she hikes 6 hours on the second day?

The equation for a car driving away from Katy is represented by the equation: 46y x , where y is the

distance from Katy after x hours.

A) What does y represent in this problem?

B) What does x represent in this problem?

C) What is the meaning of the slope?

D) What would the slope and y-intercept in the equation 46 184y x represent?

The Star Car Rental Company charges a flat fee of $30 plus $0.25 per mile to rent a car.

A) Which value would be considered the y-intercept or beginning value?

B) Which value would be considered the slope or rate of change?

C) Write the equation for this situation.

D) If You only had $50 to spend on the rental car, how many miles could you drive?


Standard Form and Intercepts

Standard Form: The equation of a line written in the form Ax + By = C A is always positive and C is a constant. x - intercept: The point where a graph touches the x – axis ( ___ , 0 ) y - intercept: The point where a graph touches the y – axis ( 0 , ___ ) Examples:

Equation: 3x + 5y = 30 x – intercept: ( , 0 ) y – intercept: ( 0 , ) Slope:

Equation: 4x – 6y = -12 x – intercept: ( , ) y – intercept: ( , ) Slope:


Practice:

Equation: 2x + 8y = -16 x – intercept: ( , ) y – intercept: ( , ) Slope:

Equation: 5x – 10y = 20 x – intercept: ( , ) y – intercept: ( , ) Slope:

Equation: 9x + 3y = 18 x – intercept: ( , ) y – intercept: ( , ) Slope:

Equation: 10x – 5y = -30 x – intercept: ( , ) y – intercept: ( , ) Slope:

Equation: 4x + 6y = -24 x – intercept: ( , ) y – intercept: ( , ) Slope:

Equation: 8x – 8y = 32 x – intercept: ( , ) y – intercept: ( , ) Slope:


Solving Standard Form for Slope Intercept Form

Examples:

Equation: 9x + 3y = 18

x – intercept: ( , )

y – intercept: ( , )

Slope:

Solve for Slope-Intercept Form: 9x + 3y = 18

Slope:



Equation: 2x + 8y = -8



Slope:

Solve for Slope-Intercept Form: 2x + 8y = -8

Slope:




Practice:

Equation: 5x – 10y = 30



Slope:

Solve for Slope-Intercept Form: 5x – 10y = 30

Slope:



Equation: 7x – 7y = 56



Slope:

Solve for Slope-Intercept Form: 7x – 7y = 56

Slope:



Continued…


Equation: 4x + 6y = -12



Slope:

Solve for Slope-Intercept Form: 4x + 6y = -12

Slope:



Equation: 10x – 5y = -40



Slope:

Solve for Slope-Intercept Form: 10x – 5y = -40

Slope:




Verifying Solutions

Solution (to a line): Any order pair that is a part of the line.

Is ( -1 , 4 ) a solution to

−2𝑦 + 2 = 6𝑥

Solve and graph

−2𝑦 + 2 = 6𝑥

Is ( 2 , -3 ) a solution to

−2𝑥 + 3𝑦 + 15 = 0

Solve and graph

−2𝑥 + 3𝑦 + 15 = 0

Is ( 4 , 2 ) a solution to

2𝑦 = 8

Solve and graph

2𝑦 = 8

Is ( -3 , 5 ) a solution to

𝑥 + 3 = 0

Solve and graph

𝑥 + 3 = 0


Practice:


−3𝑦 + 4𝑥 = 15

Solve and graph

−3𝑦 + 4𝑥 = 15

Is ( -5 , -2 ) a solution to

𝑥 + 𝑦 = 3

Solve and graph

𝑥 + 𝑦 = 3


3𝑥 + 2𝑦 − 8 = 0

Solve and graph

3𝑥 + 2𝑦 − 8 = 0


6𝑥 = −12

Solve and graph

6𝑥 = −12

Continued…


Is ( -6 , -1 ) a solution to

−2𝑥 + 6𝑦 = 6

Solve and graph

−2𝑥 + 6𝑦 = 6


4𝑦 − 3𝑥 = −8

Solve and graph

4𝑦 − 3𝑥 = −8


𝑦 − 5 = −3

Solve and graph

𝑦 − 5 = −3


−𝑥 + 𝑦 = −4

Solve and graph

−𝑥 + 𝑦 = −4

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