Monday = Point-Slope Form Investigation
Part 1:
1. Go to desmos. Start with a blank graph (no equations already listed).
2. Type in the following equations on the chart below one at a time.
a. Add the “m” slider.
b. Click on the y-intercept so you can see those values as you are manipulating the equations.
3. Move “m” either by yourself or by hitting the “play” button.
4. Watch and see that the graph is rotating around one ordered pair.
5. Type in your ordered pair prediction on line 3 in desmos. Check to see if your prediction is correct by making sure you keep
“m” moving and that your point stays on the graphed line.
Equation
Rotating Point y-intercept
Equation
Rotating Point y-intercept
A
B
y  3  m( x  2)
y  5  m( x  1)
E
F
y  4  m( x  5)
y  2  m( x  6)
C
y  1  m( x  3)
G
y  6  m( x  2)
D
y  2  m( x  4)
H
y  3  m( x  5)
6. What does “m” stand for in this equation?
7. Do you notice a pattern with the rotating point and the values in the equation? How do you think you can predict the rotating
point without graphing the equation? Write an example if that will help you.
8. The rotating point is actually the starting point for point-slope form.
a. Is this the same starting point as slope-intercept form?
b. If the equation of the line is written in point-slope form, do you start by graphing the y-intercept?
Part 2:
1. Each question has two equations. You will transition from part 1 to being able to graph in point-slope form on your own. Pay
attention to the minimal difference between the equations.
2. Type in the top equation. Figure out your starting point “rotating point”. If you need to type it in the third equation line to
check, you may do so. Write your starting point on the line below the equations.
3. Now, look at the bottom equation. Move the slider so that the slope on desmos is the same as the number in front of the
parenthesis. The number in front is your slope (“m”).
4. Graph your slope from the starting point. You need to graph at least 3 points.
5. On third line, type in the second equation. Make sure that both desmos graphs and your graph is the same.
1.
y  1  m( x  5)
y  1  2( x  5)
y  5  m  x  4
y  2  m  x  3
2.
y2
1
 x  3
2
3.
y 5 
Starting pt. =
Starting pt. =
Starting pt. =
Slope =
Slope =
Slope =
y-intercept =
y-intercept =
y-intercept =
2
 x  4
3
1
y  4  m( x  1)
y  4  ( x  1)
4.
5.
y  5  m  x  2
y  5   x  2
y  4  m  x  2
6.
y4 
Starting pt. =
Starting pt. =
Starting pt. =
Slope =
Slope =
Slope =
y-intercept =
y-intercept =
y-intercept =
1
 x  2
3
Part 3: Point-Slope Simulation
1.
Explore the simulation for point-slope form. Look at how the purple ordered pair changes the equation. What
do you notice?
2.
Try and make a line with zero slope. Then try and make a line with an undefined slope. What happened to the
equation?
3.
Look to put different points on the line with the grey boxes at the bottom of the screen.
4.
Now, write the following equations either using the simulator, desmos, or your knowledge in point-slope form
given the following information.
A.
Passes through (2, 3) and has a slope of -1/2.
B.
Passes through (-1, 4) and m = 4.
C.
Passes through (-4, 6) and (-2, 5).
D.
Passes through (-1, -7) and (1, 3).
Summary: Point-Slope form is
y  y1  m  x  x1 
1.
Why do you think it is called point-slope form (remember math people are not creative)? In other words,
what information do you think you can find in point-slope form?
2.
Why do you think it is called slope-intercept form (remember math people are not creative)? In other
words, what information do you think you can find in slope-intercept form?
3.
What do you think the letters in point-slope form represent?
y1 
m
a. x1 
b.
c.
State how you would graph a line in point-slope form on your own without demos.
4.
2
HW for Monday = Watch and take notes on the following equations in Point-Slope Form
y2 
1.
1
 x  4
2
2.
y 1 
2
 x  3
3
y  2   x  4
3.
Starting pt. =
Starting pt. =
Starting pt. =
Slope =
Slope =
Slope =
4.
y  4    x  1
5.
y  2  x  3
y  4  3( x  2)
3.
Starting pt. =
Starting pt. =
Starting pt. =
Slope =
Slope =
Slope =
Graph
Slope values
Possible Equation
Zero Slope
Positive Slope
Negative Slope
Undefined
3
Tuesday = Point-Slope Day 2 (HW = finish all PS and work on Logo project)
Point-Slope Practice from PPT – what is the point-slope form:
2. y  5  3( x  2)
1
1. y  3 
 x  4
2
3. y  4 
2
 x  2
3
4. y  5 
1
 x  2
2
5. y  5 
2
 x  2
3
6. y  1 
1
 x  2
3
7. y  2 
3
 x  3
4
8. y  4 
4
 x  2
5
9. y  2 
5
 x  2
2
Practice Part 2:
4
Practice Part 3:
5
Wednesday = Standard Form Investigation
Part 1:
1. Go to demsos. Clear all other equations.
a. Type in ax  by  c into the first equation.
b. Click to add all sliders.
c. Change your range to go from -20 to 20 for all sliders. Change step to 1.
d. Click on the x- and y-intercepts so you will see those ordered pairs at all times.
2. Press “play” on the “a” slider ONLY.
a. What happens to the slope?
b. What happens to the y-intercept?
c. What happens to the x-intercept?
3. “Pause” the “a” slider. Press “play” on the “b” slider ONLY. It does not matter where you stopped the “a” slider.
a. What happens to the slope?
b. What happens to the y-intercept?
c. What happens to the x-intercept?
4. “Pause” the “b” slider. Press “play” on the “c” slider ONLY. It does not matter where you stopped the “b” slider.
a. What happens to the slope?
b. What happens to the y-intercept?
c. What happens to the x-intercept?
5. Is the slope affected by changing the letters A, B, and/or C?
6. Is the y-intercept affected by changing the letters A, B, and/or C?
7. Is the x-intercept affected by changing the letters A, B, and/or C?
Part 2:
1. Clear out your desmos graphs. Type in y  mx  b . Click to add all sliders.
2. In the fourth line, you are going to write out the following equations in desmos. Then move the “m” and “b”
sliders until your equations match. Then answer the questions.
3. Clear your equation after you have found your slope, y-intercept, and x-intercept. Then go to the next equation.
SF form
“A”
“B”
“C”
Slope
y-intercept
x-intercept
1
3x  2 y  6
2
3x  2 y  6
3
3x  2 y  6
4
3x  2 y  6
5
x y 5
6
x y 5
3
2
6
4. What conclusions can you draw about the slope? If you look at the original equation, can you predict what the
slope will be each time?
5. What conclusions can you draw about the y-intercept? If you look at the original equation, can you predict what
the y-intercept will be each time?
6
6. Now, for the next level. Your slope, y-intercept, and x-intercept should be fractions NOT decimals.
SF form
“A”
“B”
“C”
Slope
y-intercept
x-intercept
A
2x  5 y  8
B
3x  2 y  4
C
x  2 y  1
D
2x  3 y  9
7. What conclusions can you draw from the original equations in standard form ax  by  c ?
a. How can you find the slope every time? Type your prediction into desmos as an equation on the fifth line starting
with m =
. Try playing all three letters individually, pause, and check the accuracy of your equation.
After you have finished perfecting your equation, write it below. Then cancel out the equation on desmos.
b. How can you find the y-intercept every time? Type your prediction into desmos as an ordered pair on the fifth
line. If you came up with the correct ordered pair, your point will stay on the line and on the y-axis when you play all
three letters individually, pause, and check the accuracy of your equation. After you have finished perfecting your
ordered pair, write it below. Then cancel out the ordered pair on desmos.
c. How can you find the x-intercept every time? Type your prediction into desmos as an ordered pair on the fifth
line. If you came up with the correct ordered pair, your point will stay on the line and on the x-axis when you play all
three letters individually, pause, and check the accuracy of your equation. After you have finished perfecting your
ordered pair, write it below. Then cancel out the ordered pair on desmos.
8. What do you think is a benefit of standard form?
9. List the steps of how you would graph in standard form? You should list at least two different ways.
Now, put it all together for Standard Form ax  by  c
slope =
y-intercept =
x-intercept =
Use the formulas you derived in question 7 to figure out the slope, x-intercept, and y-intercept for the
following equations in Standard Form. Check your answers in desmos.
1.
2.
3.
2 x  5 y  10
x  2 y  8
5x  y  12
slope =
slope =
slope =
y-int =
y-int =
y-int =
x-int =
x-int =
x-int =
4.
3x  5 y  8
slope =
5.
2 x  5 y  15
slope =
6.
x  4y  7
slope =
y-int =
y-int =
y-int =
x-int =
x-int =
x-int =
7
HW=Notes on Standard Form
Standard Form – Graphing
Remember, standard form is Ax + By = C!!!
a. NO FRACTIONS
b. “A” can’t be negative!
Did you know you can find the slope and y-intercept in standard form???
Standard form: Ax + By = C
A
slope =
B
 C
y-int =  0, 
 B
C 
x-int =  , 0 
A 
Remember, standard form is Ax + By = C!!! You can’t have any fractions and “A” can’t be negative!!!
1.
3x  5 y  15
2.
x  2y  8
3.
6x  y  3
slope =
slope =
slope =
y-int =
y-int =
y-int =
4.
2x  3 y  9
slope =
5.
x  6y  6
slope =
6.
3x  y  4
slope =
y-int =
y-int =
y-int =
8
Thursday = Graphing in all 3 Forms
Putting it all together in all three forms 
Slope – Intercept Form (SI):
y  mx  b
m = slope, b = y-intercept (0, b)
A
 C
Standard Form (SF): AX + BY = C slope =
, y-intercept =  0, 
B
 B
Point – Slope Form (PS): y  y1  m( x  x1 ) m = slope, starting point  x1 , y1 
Fill out the chart and then check your answers using Desmos.
y  2x 1
2 x  3 y  12
y  3  ( x  2)
4x  y  6
y 5 
1
( x  1)
2
y
3x
1
5
Form
Slope
Starting
Point
y-intercept
x-intercept
Graphing in all three forms 
You need to know what information each form gives you!
Form
1
y  2  ( x  1)
5
x
y   2
4
3x  2 y  4
SI
SI
SI
SF PS
SF PS
SF PS
Slope
Starting
Point
Graph
9
Form
Slope
SP
Graph
y  3x  2
4x  3 y  6
y  2( x  1)
SI
SI
SI
SF PS
y  3  2( x  2)
Form
Slope
SP
SI
SF PS
SF PS
x  2y  8
SI
SF PS
SF PS
y
SI
2x
4
3
SF PS
Graph
Form
y  1  ( x  5)
y  x  2
3x  y  4
SI
SI
SI
SF PS
SF PS
SF PS
Slope
SP
Graph
10
HW for Thursday = Conversion Notes
Converting from SI to SF
Don’t forget: “a” can’t be negative in standard form and NO FRACTIONS!
1.
y
1
x3
2
3
x5
4
2.
y
2.
4 x  5 y  20
2.
y2
Converting from SF to SI
1.
2 x  3 y  15
Converting from PS to SI to SF
1.
y  3  2( x  1)
All three options
1
y  3   x  4
1.
3
2.
3x  5 y  15
2
 x  5
3
3.
y
x
2
3
11
Friday = Conversion Day (whatever you do not finish is HW)
Practice Converting in Between Forms Practice Part 1:
For the following examples you are practicing converting in between the three different forms of lines. You should
check your answers on the Desmos Calculator!
If you are given SI form, convert to SF. Make sure that “A” is positive and NO fractions. For PS form, use the y-or xintercept (or any other point on the line you found via the desmos graphing website)
If you are given SF, convert to SI form. For PS form, use the x- or y-intercept (or any other point from desmos).
If you are given PS form, convert to SI form then to SF. Make sure that “A” is positive for SF and NO fractions!
Slope – Intercept Form (SI):
y  mx  b
m = slope, b = y-intercept (0, b)
A
 C
Standard Form (SF): AX + BY = C, slope =
, y-intercept =  0,  , NO fractions, “A” positive
B
 B
Point – Slope Form (PS): y 
PS
SI
y1  m( x  x1 ) m = slope, starting point  x1 , y1 
SF
y  3x  7
y  x  7
y=
1
x-2
4
y
2
x5
3
2x – y = 4
3x + 4 y = 5
x – 3y = -2
5x + 3y = - 4
y  3  2( x  5)
y  1  ( x  3)
1
y  2   ( x  3)
2
y4
2
( x  3)
3
12
Practice Part 2
#
Original Equation
Form
Slope
1
y  x  2
Starting
Point
SI
-1
(0, 2)
2
4x  3 y  9
SF
4
3
(0, 3)
3
y  4  2( x  1)
PS
2
(1, 4)
4
3x  y  5
5
y  1  ( x  2)
6
y
x
3
2
7
y
2
x2
3
8
y  4( x  3)
9
x  4y  8
10
y
11
2 x  5 y  15
12
y 3 
13
3x  5 y  15
14
y
x
2
3
15
y
x
4
2
16
y  1  2( x  3)
Point-Slope Form
y  2  1( x  0)
y 3
4
 x  0
3
y  4  2( x  1)
Slope-Intercept
Form
Standard Form
y  x  2
x y 2
4
x 3
3
y  2x  6
4x  3 y  9
y
2 x  y  6
3
x
4
1
 x  4
3
13
Graphing in all 3 forms plus conversions
2 x  3 y  6
x
y   3
3
1
y  4   ( x  3)
2
Form/
slope
S.P
Graph
SI
PS
SI
SF
SF
PS
y  2x  5
5x  4 y  8
y 3 
2
( x  5)
3
Form/
slope
SP
Graph
PS
PS
SI
SF
SI
SF
14