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Students explore the concept of slope and y-intercept in linear relationships, using function definitions as a third representation (alongside tables and graphs).

Lesson Goals

Students will be able to…​

  • define linear functions using slope and y-intercept

  • match different representations of the same linear relationship

Student-facing Lesson Goals

  • Let’s learn to connect different representations of the same linear relationship.

  • Let’s learn to identify the slope and y-intercept in a function definition.

  • Let’s define functions expressing linear relationships.

Materials

Supplemental Materials

Preparation

  • There is an optional interactive Desmos activity in the lesson. If you would like to use it, decide how you will share it with students and, if you are using our Google Slides, add the appropriate link to the slide deck. If you’re a first-time Desmos user, fear not! Here’s what you need to do.

Key Points for the Facilitator

  • Lines are made of points (try to avoid referring to "lines" as much as "collections of points")

  • Linear functions can be represented as straight lines on a graph or as sequences that change at a constant rate in a table.

  • Stress that Graphs, Definitions, and Tables are all different ways of viewing the same function. The misconception to fight against is the idea that f(x) = 3x + 2 is somehow the "real" function, with the graph being "the thing the function draws."

🔗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 sometimes the relationship between an independent variable (x) and a dependent variable (y) is linear.

We know how to identify a linear relationship represented as a table or graph. These relationships can be summarized by a function definition, but what do the definitions look like?

Linear functions are defined by their slope and y-intercept.

Here we see a function definition written using pyret notation and using function notation.

f(x) = 2x + 10. Slope = 2. Y-intercept = 10.

The slope-intercept form of the line includes the slope as the coefficient of x, and the y-intercept 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 y-intercept.

While it is common to write the x-term first and the y-intercept second, they can be written in any order!

Function Notation Pyret Code

f(x) = 6x - 10

fun f​(​x​): (​6 * x​) - 10 end

f(x) = - 10 + 6x

fun f​(​x​): -10 + (​6 * x​) end

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

f(x) = 0x + 22

f(x) = 22

When the y-intercept is 0 (and the line crosses the y-axis at the origin)…​ we may choose whether or not to write the slope term.

"Visible" y-intercept "Invisible" y-intercept

f(x) = 3.2x + 0

f(x) = 3.2x

To check our work, we can apply the function to the x-value from any coordinate pair on our table or graph, and it should produce the y-value!

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 y-intercept from function definitions.

Let’s connect definitions to tables and graphs.

Writing down the slope and y-intercept beneath each representation will help!

Let’s write our own definitions from tables and graphs!

Function definitions are a way of talking about relationships between an independent variable and a dependent variable:

  • milk costs $0.59/gallon

  • a stone falls at 9.8m/s2

  • 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.

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

What strategies did you use?

🔗Finding the y-intercept from the Slope and a Point 20 minutes

Launch

Consider the function f(x) = 3x.

x

0

1

2

3

y

0

3

6

9

  • What is the slope?

    • 3

  • What is the y-intercept?

    • 0

  • What is the y-value when x = 2?

    • 6

Anytime the y-intercept is 0, we can multiply any x-value by the slope to get its corresponding y-value.

But if the y-intercept isn’t zero…​ there is another step to finding the y-value.

Consider the function f(x) = 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 y-intercept?

    • -2

  • What is the y-value when x = 2?

    • 4 - Two less than the y-value for x = 3 in the previous function, where the y-intercept was 0.

x

0

1

2

3

y

-2

1

4

7

The y-intercept always gets added to / subtracted from the product of the slope and the x-value to find the corresponding y-value.

Investigate

We’ve learned that the relationship between the x-values and the y-values can be described using y = mx + b, where m stands for slope and b stands for the y-intercept.

If we solve that for the y-intercept…​

b = y - mx

In other words, the y-intercept can be calculated by subtracting the product of the slope and any x-value from the corresponding y-value.

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

To find the y-intercept, subtract 9 (the y-value of the point) minus 3 × 7 (the product of the slope and the x-value of the point).

  • b = 9 - 3(7)

  • b = 9 - 21

  • b = - 12…​ we found our y-intercept!

We can now use the slope and y-intercept to write our function definition:

  • f(x) = 3x - 12

Consider the table below.

x

80

81

82

83

y

150

155

160

165

  • What is the slope?

    • 5

  • Calculate the y-intercept using the first coordinate pair.

    • b = y - mx

    • b = 150 - 5(80)

    • b = 150 - 400

    • b = - 250

  • Do you get the same y-intercept if you use another pair?

    • Yes.

Synthesize

  • If we already know how to find a linear function from two points, why is it important to know how to find a linear function from the slope and just one point?

    • Sometimes we don’t always have two points!

  • What are some real-world examples of situations where we have the slope and a single point?

    • We know the rate (speed, price, etc.) and the initial value (location, startup fee, etc.), and need to figure out some point in the future.

🔗Additional Exercises

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.

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