Section 7.1 Basics of Graphing and Slope Intercept Form › uploads › 2 › 4 › 0 › 8 ›...

Foundations of Math 9 Updated January 2020

1 Adrian Herlaar, School District 61 www.mrherlaar.weebly.com

Section 7.1 – Basics of Graphing and Slope Intercept Form

This booklet belongs to: Block:

Mapping Points on a 2-D Grid

Every equation of a straight-line (except 2 special ones) has specific criteria.

They have 2 variables (unknowns), generally denoted 𝒙, 𝒚 and they have an = sign.

All lines can be mapped on a 2-D grid, called it a Cartesian plane.

The Grid is made up of 2 axes

An 𝒙 − 𝒂𝒙𝒊𝒔 and a 𝒚 − 𝒂𝒙𝒊𝒔

The axes are both number lines

The 𝑥 − 𝑎𝑥𝑖𝑠 move left and right

The 𝑦 − 𝑎𝑥𝑖𝑠 move up and down

In order to be a point found on the grid you

need both an 𝒙 − 𝒗𝒂𝒍𝒖𝒆 𝒂𝒏𝒅 𝒚 − 𝒗𝒂𝒍𝒖𝒆

denoted (𝒙, 𝒚)

Together they give the 2-D coordinates of

points on the grid

Example: See above grid for placement

(0, 0)

(1, 2)

(−4, 5)

(8, −3)

(−6, −9)

So now we know how to map out points!

Known and the ORIGIN

(1, 2)

(−4, 5)

(8, −3)

(−6, −9)

(0, 0)

Without 2 values, an 𝒙 𝒂𝒏𝒅 𝒚, it is not possible to be a

point on the grid.

Each value represents 1 dimension, and we have a

2-dimensional grid



Slope-Intercept Equation

Now think about this, a LINE is made up of an INFINTE number of individual points

There are two equations we are going to talk about this year

Here is the first one:

𝒚 = 𝒎𝒙 + 𝒃

SLOPE-INTERCEPT FORM

The variables in this equation are very important

o The 𝒎: Is the SLOPE of the line, represented 𝑅𝐼𝑆𝐸

𝑅𝑈𝑁

o 𝑅𝐼𝑆𝐸

𝑅𝑈𝑁 𝑎𝑙𝑠𝑜

𝐶𝐻𝐴𝑁𝐺𝐸 𝐼𝑁 𝐻𝐸𝐼𝐺𝐻𝑇

𝐶𝐻𝐴𝑁𝐺𝐸 𝐼𝑁 𝐿𝐸𝑁𝐺𝑇𝐻

o The Slope is the same from any point on the line to another

o The SLOPE stays constant

o The 𝒃: Is the 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕, where the line crosses the 𝑦 − 𝑎𝑥𝑖𝑠

o Why 𝑏 and not 𝑦 then?

o You will soon find out…

o Lastly, the 𝑥 𝑎𝑛𝑑 𝑦.

o They represent the (𝒙, 𝒚) coordinates of every possible point on the line

So, why 𝑏 for the 𝑦 − 𝑖𝑛𝑡?

Have a look at the grid

No matter where you cross

the 𝑦 − 𝑎𝑥𝑖𝑠, what is the 𝑥 − 𝑣𝑎𝑙𝑢𝑒?

It is always, 0

So every 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡, has the coordinates: (0, 𝑏)

The 𝒃, is wherever it crosses the 𝒚 − 𝒂𝒙𝒊𝒔.

(0, 𝑏)

No matter

where it

crosses the

𝑥 − 𝑣𝑎𝑙𝑢𝑒 is 0



Every line (except 1 type) has a 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 and has a 𝒔𝒍𝒐𝒑𝒆. As an example:

𝑦 = 𝑚𝑥 + 𝑏 → 𝑦 =2

3𝑥 + 4

The Slope (𝒎) is: 2

3 For every 2 you go up, you go right 3, both positive

Remember that the 𝑥 𝑎𝑛𝑑 𝑦 represent every set of coordinates (𝑥, 𝑦)

So the 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 is when the 𝒙 − 𝒗𝒂𝒍𝒖𝒆 is 0

We can plug 0 in to the equation for 𝑥 and then solve for 𝑦

𝑦 =2

3𝑥 + 4

𝑦 =2

3(0) + 4

𝑦 = 0 + 4

𝑦 = 4

Now let’s go back to the (𝑥, 𝑦), remember that they represent the coordinates of every possible

point on the line, they also are called the solution to the 𝑦 = 𝑚𝑥 + 𝑏 equation.

What I mean by that is that when I plug the 𝒙 𝒂𝒏𝒅 𝒚 values into the equation of a line, it

stays equal.

Example: 𝑦 =2

3𝑥 + 4

i) When 𝑥 = 0

𝑦 =2

3(0) + 4 → 𝑦 = 4 Coordinates are: (0, 4)

ii) When 𝑥 = 3

𝑦 =2

3(3) + 4 → 𝑦 = 2 + 4 = 6 Coordinates are: (3, 6)

iii) When 𝑥 = 6

𝑦 =2

3(6) + 4 → 𝑦 = 4 + 4 = 8 Coordinates are: (6, 8)

𝑦 = 𝑚𝑥 + 𝑏

𝑦 = 𝑚(0) + 𝑏

𝑦 = 0 + 𝑏

𝑦 = 𝑏

So 𝑦 = 𝑏, 𝑤ℎ𝑒𝑛 𝑥 𝑖𝑠 0

That’s why 𝑏 is the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡



We can continue this infinitely!

Pick any 𝑥 and solve for 𝑦

Pick any 𝑦 and solve for 𝑥

So, what do we know so far?

Every 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 has an 𝒙 − 𝒗𝒂𝒍𝒖𝒆 of 𝟎

Every 𝒙 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 has a 𝒚 − 𝒗𝒂𝒍𝒖𝒆 of 𝟎

We can find an infinite number of coordinates (solutions) for a line

Now for Slope we know a few things too.

The Slope of a straight line is consistent

The Slope can go up or down

The Slope is 𝑅𝑖𝑠𝑒

𝑅𝑢𝑛=

𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 ℎ𝑒𝑖𝑔ℎ𝑡

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑙𝑒𝑛𝑔𝑡ℎ=

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥

We will come across 4 different types of lines. Their characteristics will results in 4 types of Slope.

Look at them from Left to Right.

The Rise is Positive

The Run is Positive

So that means the Slope will be: 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒

𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒= 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒

The Rise is Negative

The Run is Positive

So that means the Slope will be:

𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒

𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒= 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒

The Rise is 0

The Run is Finite


0

𝐴𝑛𝑦𝑡ℎ𝑖𝑛𝑔= 0

Anything divided by 0 is 0

The Rise is Finite

The Run is 0


𝐴𝑛𝑦𝑡ℎ𝑖𝑛𝑔

0= 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

Can’t divide by zero



Solutions to a Line

Next is figuring out if a point is on a line. That is the same as saying: Is the following point

a solution to the equation of the line.

If the point is a solution, then when you plug the (𝑥, 𝑦) into the given equation, it will

stay equal, and the point is on the line

If the point is not a solution, then when you plug the (𝑥, 𝑦) into the given equation, it

will not stay equal, and the point is not on the line

Example: Does the line 𝑦 = 2𝑥 + 5 go through the point (1,8)?

Solution:

Since 𝒙 is 1, we plug 1 in for 𝒙 and since 𝒚 is 8, we plug 8 in for 𝒚.

Work through the equation and see if it stays equal.

If it does, it’s a solution (A point on the line)

If it doesn’t, it’s not a solution (Not a point on the line)

𝑦 = 2𝑥 + 5

8 = 2(1) + 5

8 = 2 + 5

8 = 7

8 DOES NOT EQUAL 7

So that means that (1, 8) is NOT a solution to 𝑦 = 2𝑥 + 5

In other words, the point at (1, 8) is not on the line with the equation 𝑦 = 2𝑥 + 5

Example:

Does the line 𝑦 = −2

5𝑥 + 6 go through the point (10, 2)?

𝑦 = −2

5(10) + 6 → 2 = −

2

5(10) + 6

2 = −20

5+ 6 → 2 = −4 + 6

𝟐 = 𝟐

So that means, the point (10, 2) is a point on the line.



Writing the Equation of a Line

We can also identify information in a graph that will allow us to write the equation of a line.

This technique is limited to SLOPE-INTERCEPT FORM and graphs where the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 is

easily discernible.

What is the equation of the given line?

o Identify the Slope and the 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 and you’re done

Try another one:

The 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 is easy to see: (𝟎, 𝟓)

Now from left to right, count the SLOPE

Our RUN: We move 7 places in the positive direction

Our RISE: We move 5 places up in the positive direction

So the SLOPE is: 𝟓

𝟕

The Equation of the line then is:

𝑦 =5

7𝑥 + 5

The 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 is easy to see: (𝟎, 𝟒)

Now from left to right, count the SLOPE

Our RUN: We move 8 places in the positive direction

Our RISE: We move 4 places up in the negative direction

So the SLOPE is: −𝟒

𝟖=

−𝟏

𝟐= −

𝟏

𝟐

The Equation of the line then is:

𝑦 = −1

2𝑥 + 4



Graphing Lines

With the SLOPE-INTERCEPT equation it is pretty easy to graph lines too.

We are given the SLOPE and the Y-INTERCEPT, so it is really quite simple.

Identify the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 from the equation and plot it

Then from that point, count out your SLOPE

Up and left, up and right, down and left, or down and right

Graph this: 𝑦 =5

3𝑥 − 2

Let’s try a couple more:

Graph: 𝑦 = −2𝑥 + 1 Graph: 𝑦 =3

4𝑥 − 5

𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: (0, −2)

𝑆𝑙𝑜𝑝𝑒: 𝑅𝑖𝑠𝑒

𝑅𝑢𝑛=

5

3

𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: (0, 1)


𝑅𝑢𝑛=

−2

1

𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: (0, −5)


𝑅𝑢𝑛=

3

4



Vertical and Horizontal Lines

Horizontal Lines

Let’s look at an example:

What is the Slope?

What is the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡?

o So the Slope is 0, and the 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 is 5.

o But when else is 𝑦 = 5?

o Does it matter what the 𝑥 − 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠?

o So do we even need 𝑥 in our equation?

It turn out that every horizontal line is simply:

𝒚 = 𝒃

So in this case, the equation of the horizontal line is:

𝒚 = 𝟓

Vertical Lines

Vertical lines don’t have the same 𝑦 − 𝑣𝑎𝑙𝑢𝑒 all the time, they have the same 𝑥 − 𝑣𝑎𝑙𝑢𝑒

So does the 𝑦 − 𝑣𝑎𝑙𝑢𝑒 matter?

Do we need it in our equation?

It turn out that every vertical line is simply:

𝒙 = 𝒂

So in this case, the equation of the vertical line is:

𝒙 = −𝟑



Summary

𝑦 = 𝑚𝑥 + 𝑏 Is the equation for a diagonal line (Slope-Intercept)

𝑦 = ? Is the equation of a horizontal line

𝑥 = ? Is the equation of a vertical line

𝑏 Is the value of the 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕

(𝑥, 𝑦) The coordinates of the point on a line (also the Solution to the Equation)

𝑚 Is the Slope

Is written: 𝑅𝑖𝑠𝑒

𝑅𝑢𝑛=

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 ℎ𝑒𝑖𝑔ℎ𝑡

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑙𝑒𝑛𝑔𝑡ℎ=

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥

Remember when counting out the Slope

You have a fraction so you can count 4 possible ways:

The first two give you a consistent POSITIVE SLOPE regardless of the direction you count

i) Up and to the Right (POSITIVE RISE/POSITIVE RUN) 𝐴

𝐵 which equals

𝐴

𝐵

ii) Down and to the Left (NEGATIVE RISE/NEGATIVE RUN) −𝐴

−𝐵 which equals

𝐴

𝐵

The second two give you a consistent NEGATIVE SLOPE regardless of the direction you count

iii) Down and to the Right (NEGATIVE RISE/POSITIVE RUN) −𝐴

𝐵 which equals −

𝐴

𝐵

iv) Up and the Left (POSITIVE RISE/NEGATIVE RUN) 𝐴

−𝐵 which equals −

𝐴

𝐵



Section 7.1 – Practice Questions

1. Map the following Coordinate (𝑥, 𝑦) on the 2-D plane (GRID) 𝐴(1, 3) 𝐵(9, −1) 𝐶(−4, 4) 𝐷(−7, −7) 𝐸(−5, −3) 𝐹(1,8) 𝐺(8, −2) 𝐻(−5, 2)

2. Identify the Coordinates of the given points

3. What does it mean to be a solution to an equation with respect to coordinates of a point?



4. What is the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡? What is the x-coordinate of every 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑝𝑜𝑖𝑛𝑡? Example?

5. What is the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡? What is the y-coordinate of every 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑝𝑜𝑖𝑛𝑡? Example?

6. What do you think of when you see the word SLOPE?

7. For the sake of our Math Vocabulary then:

𝑆𝐿𝑂𝑃𝐸 =

=

8. Are the following points solutions to the given equations? Are they POINTS on the given LINE? A) (1, −3) 𝑦 = 3𝑥 − 5

B) (0, 5) 𝑦 =2

3𝑥 + 5

C) (−2, 7) 𝑦 =3

2𝑥 + 4

D) (8, −1) 𝑦 = −1

8𝑥



9. What is the SLOPE and Y-INTERCEPT of the following lines?

𝑆𝑙𝑜𝑝𝑒 = 𝑆𝑙𝑜𝑝𝑒 = 𝑦 − 𝑖𝑛𝑡 = 𝑦 − 𝑖𝑛𝑡 =

10. Graph the following lines.

𝑦 = 𝑥 𝑦 =2

5𝑥 + 4

𝑎) 𝑏)



𝑦 = −2𝑥 + 7 𝑦 = −3

5𝑥 − 5

𝑦 = 3 𝑥 = −4

𝑐) 𝑑)

𝑒) 𝑓)



Answer Key – Section 7.1

1.

2.

3. It means the (𝑥, 𝑦) of a point sub into 𝑦 = 𝑚𝑥 + 𝑏 for the 𝑥 and 𝑦 and the equation stays equal

4. Where the line goes through the 𝑦 − 𝑎𝑥𝑖𝑠; 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 always 0; (0, 4)

5. Where the line goes through the 𝑥 − 𝑎𝑥𝑖𝑠; 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 always 0; (4, 0)

6. Answers Vary

7. 𝑆𝑙𝑜𝑝𝑒 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐻𝑒𝑖𝑔ℎ𝑡

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ=

𝑅𝐼𝑆𝐸

𝑅𝑈𝑁

8. A: NO B: YES C: NO D: YES

9. i) 𝑆𝑙𝑜𝑝𝑒 𝑖𝑠 3

4, 𝑦 − 𝑖𝑛𝑡 𝑖𝑠 (0, 3) ii) 𝑆𝑙𝑜𝑝𝑒 𝑖𝑠 −

1

7, 𝑦 − 𝑖𝑛𝑡 𝑖𝑠 (0, 4)

10.

𝑦 = 𝑥 𝑦 =2

5𝑥 + 4

𝐴

𝐵

𝐷

C

𝐸

𝐹

𝐺

𝐻

(−7, 7)

(0,6)

(4, 5)

(0, 0) (7, 0)

(−9, −1)

(−6, −5)

(6, − 7)

𝑎) 𝑏)



𝑦 = −2𝑥 + 7 𝑦 = −

3

5𝑥 − 5

𝑦 = 3 𝑥 = −4

𝑐) 𝑑)

𝑒) 𝑓)



Extra Work Space

Section 7.1 Basics of Graphing and Slope Intercept Form › uploads › 2 › 4 › 0 › 8 ›...

Documents

Transcript of Section 7.1 Basics of Graphing and Slope Intercept Form › uploads › 2 › 4 › 0 › 8 ›...