Key Features of Functions – Mathberry Lane

Category: Key Features of Functions

14
Sep
2025

How to teach X and Y Intercepts with a A Hands On, Visual Approach: Algebra Key Features of Functions,

categories: Free Resources, Key Features of Functions

X- and y-intercepts are one of those foundational concepts that pop up again and again in algebra, graphing, and real-world modeling. But for students, they can feel abstract—just another pair of coordinates to memorize. So how do we make intercepts click?

Here’s a step-by-step approach that blends visual learning, student agency, and collaborative thinking to help students not just find intercepts—but understand what they mean.

🧠 Step 1: Start with Noticing

Begin with a warm-up that invites curiosity. Show a couple of graphs and ask students:
“What do you notice?”
Let them pair-share, jot ideas on sticky notes, or do a silent chalk talk. This primes their brains to look for patterns and builds confidence before any formal instruction begins

🎨 Step 2: Color the Axes

Hand out colored pencils or highlighters and have students trace the x-axis one color and the y-axis another. This simple move helps anchor their spatial understanding and makes it easier to spot intercepts visually.

Then ask:

Where does the graph cross the x-axis?
Where does it cross the y-axis?

Let students circle those points and label them. You’re building intuitive understanding before introducing vocabulary.

📊 Step 3: Explore Tables and Graphs

Give students graphs and tables of functions. Ask them to find the intercepts in each format. Then flip the task: give them intercepts and ask them to sketch possible graphs. This back-and-forth builds flexibility and reinforces the idea that intercepts are where the function meets the axes—not just numbers to plug in.

🧩 Step 4: Define and Organize

Now that students have seen intercepts in action, introduce formal definitions. Use a graphic organizer to show:

How to find intercepts from a graph
How to find them from a table
How to find them from an equation

This organizer becomes a reference tool they can return to throughout the unit.

✏️ Step 5: Practice with Purpose

Use practice problems that ask students to:

Identify intercepts from different representations
Create graphs with given intercepts
Match equations to intercepts

For early finishers, offer an extension:
“Can you make a table for this graph?” or
“Can you graph this table?”
This keeps students engaged and deepens their understanding of how intercepts connect across formats.

📎 Want a Ready-to-Go Resource?

If you’d like a lesson that walks students through all of this—complete with warm-ups, notes, graphic organizers, and practice pages—I’ve got one ready for you. It’s available on my TpT store, and it’s designed to be flexible, visual, and student-friendly.

Grab this X and Y Intercept Graphic Organizer!

Get Your Free Copy Here!

Whether you’re introducing intercepts for the first time or revisiting them before diving into linear equations, this approach helps students build lasting understanding—one axis at a time.

Thanks for stopping by! I hope this lesson idea brings a little more clarity (and color!) to your classroom. And if you grabbed the free graphic organizer, keep an eye on your inbox—I’ve got more teaching inspiration headed your way soon.

01
Aug
2023

Teaching Increasing and Decreasing Functions

categories: Key Features of Functions, Mathberry Lane Resources on TPT

For some reason students tend to have difficulty with this concept. At first glance, it seems rather straightforward. If a graph is going up, then the function is increasing. If the graph is going down, then the function is decreasing. If the graph is a horizontal line, then the function is constant. This is the level of this key feature of functions taught in my algebra 2 class.

In calculus, we take it a step further and look at intervals where the first derivative is positive to see where the function is increasing. And likewise, intervals where the first derivative is negative tells us where the function is decreasing. If the first derivative equals zero, then the function is constant.

Next, are the intervals for increasing and decreasing open or closed? Different textbooks will have different notations. Some will use open intervals while others will use closed. I used to be in the open interval camp, but have switched over to closed. I discuss this with my students about how textbook publishers and even math teachers cannot agree on this topic. With that said, if a function is increasing as it goes to an asymptote or infinity, then an open interval would be used.

Let’s look at the definition for a function increasing or decreasing on an interval.

I start this topic with a what do you notice prompt for the graph shown above. It is an entry point into the lesson where each student can share what they see. We then get out some crayons or colored pencils to shade in different portions of the graph.

Next I ask students to try sketching their own examples of graphs that are increasing, decreasing or constant, followed by a graph that exhibits more than one of these behaviors.

Before we start creating intervals for increasing and decreasing, we first review interval notation. Click HERE if you are interested in my free graphic organizer for interval notation.

Finally, we analyze graphs to identify intervals of increasing, decreasing or constant. After students feel comfortable identifying these intervals, I give them intervals with specified criteria and ask them to create the graph that would meet these criteria.

If you are interested in my lesson, it is available in my TPT shop as part of my Key Features of Functions Unit. I have included 2 versions: closed interval notation and open interval notation. Cheers!

02
Jul
2023

Teaching Interval Notation

categories: Key Features of Functions

Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value. Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.

Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead. When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.

Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.

As an inequality we would write this as x ≤ 2. However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.

Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.

As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).

We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound inequality.

The set of all real numbers can also be written with interval notation.

A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).

You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.