Graphing Equations in Slope Intercept Form
Let’s remember a linear equation can be written in:
Standard Form: Ax + By = C
It can also be written as:
Slope-Intercept Form: y = mx + b (m = slope and b = y-intercept)
To Graph:
1. Plot the y-intercept (b in the equation).
2. From the y-intercept, use the slope to tell us how to move:
Example: m= → means from y-int. move UP 3 units and RIGHT 4 units.
m= → means from y-int. move DOWN 2 units and RIGHT 3 units
It is much easier to graph a line when the equation is in Slope-Intercept form.
Write an equation in slope-intercept form for a line with a slope of and a y-
intercept of -2. Then graph.
Plug in what you know:
y = mx + b
𝑦 = 𝑥 − 2
y-intercept = -2
Slope means up 3 right 4
Write an equation in slope-intercept form for the line shown in the graph.
To write an equation in slope-intercept form we need the slope and y-intercept.
(Hint: you will need to use the slope formula and two points on the line to
calculate slope).
In the above graph we see that the y-intercept = 1 or (0, 1).
Pick another point on the line (in this case, (2,0)).
Calculate slope using these two points: 𝑚 =0−1
2−0=
−1
2
Put it all together:
𝑦 = −1
2 𝑥 + 1
EXAMPLE: The band boosters are selling sandwiches for $5 each. They bought
$1160 worth of ingredients. Write an equation for the profit made on x
sandwiches and graph.
𝑦 =5𝑥−1160
It’s important to know that x cannot be negative, since x represents number of sandwiches.
What would the profit be if 1400 sandwiches are sold (x=1400).
𝑦=5(1400)−1160
𝑦 =7000−1160
𝑦 =5840
The profit from selling 1400 sandwiches would be $5,840.
- 1160
Writing Equations in Slope-Intercept Form
Slope Intercept Form: y=mx + b (m = slope, b = y-int)
If we are given:
We can:
Slope and coordinate
A line has a slope of 1
2 and
passes through the point
(2, 4)
We can use the slope and coordinate to find y-int. (b):
𝑦 =𝑚𝑥+𝑏 Substitute m, x and y (from coordinate) and solve for b.
4=(2)(1
2)+𝑏
4=1+𝑏
3=𝑏
Now write the equation:
𝑦 =1
2𝑥+3
Two coordinate points
A line passes through the
points (1, 2) and (3, 5)
1. Calculate slope using slope formula
𝑚 =5 − 2
3 − 1=
3
2
2. Now that we have the slope, we can use the procedure from above to find b. Just use one of the given points:
2=(1)(3
2)+𝑏
1
2= 𝑏
3. Write the equation 𝑦 =3
2𝑥 +
1
2
You Try! Write equations in slope-intercept form using the given information:
Slope = -
2
3 and passes through (3, 1)
𝑦 = −2
3𝑥 + 𝑏
1 = −2
3(3) + 𝑏
1 = −2 + 𝑏
3 = 𝑏
𝑦 = −2
3𝑥 + 3
Slope = 5 and passes through (0, 0)
𝑦 = 5𝑥 + 𝑏
0 = 5(0) + 𝑏
0 = 𝑏
𝑦 = 5𝑥
Line that passes through (-3, 1) and (4, 5)
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑚 =
5 − 1
4 + 3=
4
7
1 =
4
7(−3) + 𝑏
1 =
−12
7+ 𝑏
2 5
7= 𝑏
𝑦 =
4
7𝑥 + 2
5
7
Line that passes through (0, -4) and (5, 5)
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑚 =
5 + 4
5 − 0=
9
5
−4 =
9
5(0) + 𝑏
−4 = 𝑏
𝑦 =9
5𝑥 − 4
I can write linear equations in slope-intercept form given…..
The slope and y-intercept
Slope = 𝟏
𝟑 y-int = -5
𝒚 = 𝟏
𝟑𝒙 − 𝟓
Standard Form
4x – 2y = 14
𝟒𝒙 − 𝟐𝒚 = 𝟏𝟒
-𝟒𝒙 − 𝟒𝒙 −𝟐𝒚 = −𝟒𝒙 + 𝟏𝟒
𝒚 = 𝟐𝒙 − 𝟕
A graph
m = −𝟐
𝟓 b = -1
𝒚 =
−𝟐
𝟓𝒙 − 𝟏
A point and a slope
(-1, 3) and slope = -3
𝒚 = 𝒎𝒙 + 𝒃
𝟑 = −𝟑(−𝟏) + 𝒃 𝟑 = 𝟑 + 𝒃
𝒃 = 𝟎 𝒚 = −𝟑𝒙
Two points
(-4, -7) and (8, -13)
𝒎 =−𝟕 + 𝟏𝟑
−𝟒 − 𝟖=
𝟔
−𝟏𝟐=
−𝟏
𝟐
𝒚 = 𝒎𝒙 + 𝒃
−𝟕 = −𝟏
𝟐(−𝟒) + 𝒃
−𝟕 = 𝟐 + 𝒃 𝒃 = −𝟗
𝒚 = −𝟏
𝟐𝒙 − 𝟗
Writing Equations in Point-Slope Form
Remember:
We’ve learned how to write linear equations in:
Standard Form: Ax + By = C
And
Slope-Intercept Form: y = mx + b
Today let’s write equations in Point-Slope Form. We use this form
when we are given the slope of the line and one coordinate on the
line.
Point-Slope Form: 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)
where 𝑥1 𝑎𝑛𝑑 𝑦1 are given coordinates
EXAMPLE:
Write an equation in point-slope form for the line that passes through (3, -
2) with a slope of .
𝒚 − (−𝟐) =1
4 (𝒙 − 𝟑)
𝒚 + 𝟐 =1
4 (𝒙 − 𝟑)
NOTE: If given two points, use slope formula to get slope. Then use
the slope and either of the given coordinates to write the equation
in point-slope form.
Converting linear equations from one form to another:
To convert from one form to another, use the rules of algebra to make
the equation reflect the new form.
To convert to Standard Form (Ax + By = C):
Goal is to get the x term and y term on one side of the equal sign and
the constant on the other. Remember the rules for A, B and C!
To convert to Slope-Intercept Form (y = mx + b):
Goal is to get the y alone on one side of the equal sign.
YOU TRY!
Convert to Standard Form:
𝑦 − 1 = − (𝑥 − 5)
𝑦 − 1 =−2
3𝑥 +
10
3
2
3𝑥 + 𝑦 =
13
3
2𝑥 + 3𝑦 = 13
Convert to Slope-Int Form:
𝑦 + 3 = (𝑥 + 1)
𝑦 + 3 =3
2𝑥 +
3
2
𝑦 =3
2𝑥 −
3
2
Convert to Standard Form:
𝑦 =1
2 𝑥 − 3
−1
2𝑥 + 𝑦 = −3
𝑥 − 2𝑦 = 6
Convert to Slope-Int Form:
3𝑥 + 4𝑦 = 12
4𝑦 = −3𝑥 + 12
𝑦 = −3
4𝑥 + 3
Parallel and Perpendicular Lines
Parallel lines: lines that never meet.
Perpendicular Lines: lines that cross and form right angles
NOTE: Since both of these consist of two lines, there are two equations.
Lines Rule Examples
Parallel
• The two lines have the SAME slope.
• The two lines have DIFFERENT y – intercepts.
𝑦=2𝑥+4
𝑦=2𝑥−3 𝑦=2𝑥
All have slope of 2
Perpendicular
• The two lines have slopes
that are OPPOSITE RECIPROCALS.
• Any y-intercepts.
𝑦=−3𝑥+6
𝑦 =1
3𝑥 − 2
-3 and opposite reciprocals
Intersecting
• There is NO relationship
between the slopes.
• Any y-intercepts.
𝑦=4𝑥+2 𝑦=−4𝑥−5
4 and -4 are not the same and they are not opposite reciprocals
It is easiest to work with equations in slope-intercept form
Steps in determining parallel and perpendicular lines or writing
equations of parallel and perpendicular lines:
Step 1: Convert the equation of the given line to slope-int form.
Step 2: Determine slope of the new line.
Step 3: Plug in what you know in y = mx + b to find the value of
b Step 4: Write the equation of the new line.
YOU TRY! Write an equation for the line that passes through (-3, 5) and is parallel to y = 2x – 4.
Slope of given line = 2 therefore
The slope of the second line = 2 (parallel- same) Find y-int (b) by using slope and coordinate:
5 = 2(−3) + 𝑏 5 = −6 + 𝑏
11 = 𝑏 Now write equation:
𝑦 = 2𝑥 + 11
Write the equation of the line that passes through (-4, 6) and is perpendicular to 2x + 3y = 12. First we need to put the first equation in slope intercept form:
3𝑦 = −2𝑥 + 12
𝑦 = −2
3𝑥 + 4
Now we can see that the slope for the
perpendicular line is 3
2 (opposite reciprocal
of −2
3 )
Find y-int by using slope and coordinate:
6 = (−4)(3
2) + 𝑏
6 = −6 + 𝑏 12 = 𝑏
Now write equation: 𝑦 = 3
2𝑥 + 12
We can prove two or more lines are parallel or perpendicular to each
other just by looking at the slopes of the lines.
Determine whether these lines are parallel or perpendicular to each
other:
Looking at the slopes we see that Line 2 and Line 3 are parallel,
Line 1 and Line 2 are perpendicular and
Line 1 and Line 3 are perpendicular
Line 1: 3x + y = 12
Line 2: y = 1
3 x + 2
Line 3: 2x – y = 6 - 5
Line1:
3 𝑥 + 𝑦 = 12
𝑦 = − 3 𝑥 + 12
𝑠𝑙𝑜𝑝𝑒 = − 3
Line 2:
𝑦 = 1
3 𝑥 + 2
𝑠𝑙𝑜𝑝𝑒 = 1
3
Line 3:
2 𝑥 − 6 𝑦 = − 5
− 6 𝑦 = − 2 𝑥 − 5
𝑦 = − 2
− 6 𝑥 −
5
6
𝑦 = 1
3 𝑥 −
5
6
𝑠𝑙𝑜𝑝𝑒 = 1
3
First step is to
convert all to
slope - int form
Parallel and Perpendicular
Use the graph to answer the following two questions:
A (2, 4) B(3, 6) C (6, 4) D (5, 2)
1. Is angle B a right angle? METHOD TO SOLVE: To answer this question, we need to determine if line AB is perpendicular to line BC. Use the slope formula to find slope of both lines – if they are opposite reciprocals, the we have perpendicular lines which makes angle B a right angle.
2. Is each pair of opposite sides
parallel? METHOD TO SOLVE: Find the slope of all four lines using slope formula. If opposite sides have equal slopes then they are parallel.
A
B
C
D
Scatter Plots and Line of Best Fit
Bivariate data Data with two variables
Scatter Plot Shows the relationship between a set of data
with two variables. The data is graphed as
ordered pairs on a coordinate plane.
Line of fit (trend) Line that models a trend within the data.
Examples of Bivariate Data:
The number of hours worked and wage earned
The number of hours studying and grades.
Can you name some other examples?
Scatter Plots:
Positive slope Negative Slope No
Positive Correlation Negative Correlation Correlation
We can draw and write an equation for a line that shows the trend
of the data ( Line of Best Fit ).
Step 1: Make scatter plot & determine relationship
Step 2: Draw a line that passes close to most of the data
Step 3: Use two points on the line you drew (not
necessarily part of the data) to write an equation.
Step 4: Use that line to make predictions.
I will use points (16, 300) and (22, 500):
𝑚 =500−300
22−16=
200
6=
100
3
𝑦 = 𝑚𝑥 + 𝑏
300 = (16) + 𝑏
300 = 533.33 + 𝑏
−233.33 = 𝑏
Line of fit equation is: 𝑦 = 𝑥 − 233.33
Inverse Linear Functions
Inverse Relation The set of ordered pairs that we get by
switching the x-coordinate and y-coordinate.
Inverse Relations
Relation A Relation B
(-3, -16) (-16, -3)
(-1, 4) (4, -1)
(2, 14) (14, 2)
(5, 32) (32, 5)
YOU TRY:
Identify the Inverse Relation:
Relation: (3, 5), (4, 6), (5, 7)
Inverse: (5, 3), (6, 4), (7, 5)
Relation: (-1, -1), (2, 3), (4, 7)
Inverse: (-1, -1), (3, 2), (7, 4)
Relation: (.1, 5), (.2, 6), (.3, 7)
Inverse: (5, .1), (6, .2), (7, .3)
Relation: (8, 4), (6, 3), (4, 2)
Inverse: (4, 8), (3, 6), (2, 4)
Switch x and y values
Function Symbols: f(x) read as function of x (the output)
𝒇−𝟏(𝒙) read as inverse of f(x)
Steps to finding the inverse of a function:
Step 1: Replace f(x) with y
Step 2: Switch x and y
Step 3: Isolate y (get y alone)
Step 4: Replace y with 𝑓−1(𝑥)
YOU TRY! a) f(x) = 4x - 8 b) f(x) = x + 11
Step 1:
Step 2:
Step 3:
Step 4:
𝑦 = 4𝑥 − 8
𝑥 = 4𝑦 − 8
4𝑦 = 𝑥 + 8 𝑦
= 𝑥 + 2
1
𝑓−1 = 𝑥 + 2
4
𝑦 = − 𝑥 + 11
𝑥 = − 𝑦 + 11
𝑦 = −𝑥 + 11
𝑦 = −2𝑥 + 22
𝑓−1 = −2𝑥 + 22
