Standards in this Lesson 

Practices in this Lesson 

(Also available in WeScheme)
Students explore the concept of slope and yintercept in linear relationships, using function definitions as a third representation (alongside tables and graphs).
Lesson Goals 
Students will be able to…

Studentfacing Lesson Goals 

Materials 

Key Points for the Facilitator 

 coordinate pair

a set of numbers describing an object’s location on the coordinate plane
 domain

the type or set of inputs that a function expects
 range

the type or set of outputs that a function produces
 slope

the steepness of a straight line on a graph
 yintercept

the point where a line or curve crosses the yaxis of a graph
🔗Defining Linear Functions 35 minutes
Overview
Students explore function definitions as a way of expressing linear relationships, and construct tables and graphs from those definitions.
Launch
As you’ve seen, a function definition is a way of summarizing a relationship. You’ve seen how a linear relationship can be expressed as a table or graph. But what do these kinds of relationships look like as a definition?
Linear functions are defined by their slope and yintercept
Here we see a function definition written using pyret notation and using function notation.
The slopeintercept form of the line includes the slope as the coefficient of x and the yintercept as the numerical term. You will hear people describe this form as y = mx + b, where m stands for slope and b stands for the yintercept.
While it is common to write the xterm first and the yintercept second, they can be written in any order!
Function Notation  Pyret Code 

fx = 6x  10 

fx = 10 + 6x 

When the slope is zero (and the line is horizontal)… we may choose whether or not to write the slope term.
"Visible" Slope  "Invisible" Slope 

fx = 0x + 22 
fx = 22 
When the yintercept is 0 (and the line crosses the yaxis at the origin)… we may choose whether or not to write the slope term.
"Visible" yintercept  "Invisible" yintercept 

fx = 3.2x + 0 
fx = 3.2x 
To check our work, we can apply the function to the xvalue from any coordinate pair on our table or graph, and it should produce the yvalue!
As with tables and graphs, a function definition can also reveal whether or not the function is linear. Functions that are not linear will follow other forms, for example they may include exponents or absolute values.
Investigate
Let’s start by identifying the slope and yintercept from function definitions.
More practice is available through our Desmos card sort activities:
Students can identify slope and yintercept from functions in Pyret as well, using Exploring Linearity in Definitions Starter File.
Let’s connect definitions to tables and graphs.
Writing down the slope and yintercept beneath each representation will help!
More matching practice is available through our Desmos card sort activities.
Let’s write our own definitions from tables and graphs!
What strategies did you use?
Students can practice defining linear functions when given tables and graphs using the Exploring Linearity in Tables Starter File and Exploring Linearity in Graphs Starter File.
Common Misconceptions
It is common to think of the graph as the "output" of the function, rather than the function itself. Most math textbooks will use language like "matching the graph to the function", suggesting that the graph is somehow not the function! Since this language is pervasive, it’s important to actively push against it.
Synthesize
Function definitions are a way of talking about relationships between quantities: milk costs $0.59/gallon, a stone falls at 9.8m/s^2, or there are 30 students for every teacher at a school. If we can figure out the relationship between a small sample of data, we can make predictions about what happens next. We can see these relationships as tables, graphs, or symbols in a definition. We can even think about them as a mapping between Domain and Range!
When we talk about functions, it’s helpful to be able to switch between representations, and see the connections between them.
🔗Finding the yintercept from the Slope and a Point 20 minutes
Launch
Consider the function fx = 3x.
x 
0 
1 
2 
3 
y 
0 
3 
6 
9 

What is the slope?

3


What is the yintercept?

0


What is the yvalue when x = 2?

6

Anytime the yintercept is 0, we can multiply any xvalue by the slope to get its corresponding yvalue.
But if the yintercept isn’t zero… there is another step to finding the yvalue.
Consider the function fx = 3x  2.
x 
0 
1 
2 
3 
y 
2 
1 
4 
7 

What is the slope?

3. Same as for the previous function.


What is the yintercept?

2


What is the yvalue when x = 2?

4. Two less than the yvalue for x=3 in the previous function, where the yintercept was 0.

The yintercept always gets added to / subtracted from the product of the slope and the xvalue to find the corresponding yvalue.
Investigate
As discussed above, the relationship between the xvalues and the yvalues can be described using y = mx + b, where m stands for slope and b stands for the yintercept.
If we solve that for the yintercept…
b = y  mx
In other words, the yintercept can be calculated by subtracting the product of the slope and any xvalue from the corrseponding yvalue.
Let’s say the slope is 3. And we know that the line passes through the point (7,9).

b = y  mx

m = 3

x = 7

y = 9

so… b = 9  37
To find the yintercept, subtract 9 (the yvalue of the point) minus 3 × 7 (the product of the slope and the xvalue of the point).
9  21 = 12
yintercept: 12
function definition making use of the yintercept we found: fx = 3x  12
Consider the table below.
x 
80 
81 
82 
83 
y 
150 
155 
160 
165 

What is the slope?

5


Calculate the yintercept using the first coordinate pair.

250


Do you get the same yintercept if you use another pair?

Yes.

🔗Additional Exercises:

Matching Graphs & Definitions of Functions (not just linear!) (Desmos)

Identifying yintercepts in Tables, Graphs & Definitions of Linear Functions (challenge) (Desmos)

Identifying Linearity in Tables, Graphs & Definitions of Linear Functions (Desmos)

Matching Tables, Graphs, and Definitions of Functions (challenge!) (Desmos)

Matching Tables, Graphs, and Definitions of Functions (challenge!) (Desmos)

Identifying yintercepts in Tables, Graphs & Definitions of Linear Functions (challenge!) (Desmos)
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