Post on 07-Jun-2020
Foundations of Math 9 Updated January 2020
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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
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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
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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
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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
So that means the Slope will be:
0
𝐴𝑛𝑦𝑡ℎ𝑖𝑛𝑔= 0
Anything divided by 0 is 0
The Rise is Finite
The Run is 0
So that means the Slope will be:
𝐴𝑛𝑦𝑡ℎ𝑖𝑛𝑔
0= 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Can’t divide by zero
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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.
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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
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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)
𝑆𝑙𝑜𝑝𝑒: 𝑅𝑖𝑠𝑒
𝑅𝑢𝑛=
−2
1
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: (0, −5)
𝑆𝑙𝑜𝑝𝑒: 𝑅𝑖𝑠𝑒
𝑅𝑢𝑛=
3
4
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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
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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
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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?
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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
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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)
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𝑦 = −2𝑥 + 7 𝑦 = −3
5𝑥 − 5
𝑦 = 3 𝑥 = −4
𝑐) 𝑑)
𝑒) 𝑓)
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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)
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𝑦 = −2𝑥 + 7 𝑦 = −
3
5𝑥 − 5
𝑦 = 3 𝑥 = −4
𝑐) 𝑑)
𝑒) 𝑓)
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Extra Work Space