Lesson Plan
Is it a Function?
Students will be able to define functions and relations, and accurately apply the vertical line test to distinguish between them using various graphical representations.
Understanding functions is fundamental to advanced mathematics. This lesson provides a crucial visual tool to identify functions, simplifying complex concepts and enabling students to interpret graphs effectively in real-world contexts.
Audience
9th Grade Students
Time
65 minutes
Approach
Direct instruction, guided practice, and independent application.
Materials
Whiteboard or projector, Slide Deck: Is it a Function?, Markers/Pens, Vertical Line Test Sort Activity, Scissors, Glue or tape (optional), Graph Analysis Worksheet, and Exit Ticket: Function Check-Up
Prep
Teacher Preparation
20 minutes
- Review the Slide Deck: Is it a Function? and familiarize yourself with the content.
- Print and cut out the cards for the Vertical Line Test Sort Activity (one set per small group or pair).
- Make copies of the Graph Analysis Worksheet (one per student).
- Make copies of the Exit Ticket: Function Check-Up (one per student).
- Ensure projector/whiteboard is set up and ready.
Step 1
Introduction & Hook: What's the Rule?
10 minutes
- Display a simple relation (e.g., a set of coordinate pairs or a basic graph) that is a function and one that is not a function, without labeling them.
- Ask students: 'What do you notice about these two relationships? Do they seem different in any way?' (5 minutes)
- Introduce the term 'function' as a special type of relationship where each input has exactly one output.
- Briefly explain that today's lesson will give them a powerful tool to identify functions quickly.
Step 2
Direct Instruction: The Vertical Line Test Unveiled
15 minutes
- Use the Slide Deck: Is it a Function? to introduce key vocabulary: relation, function, domain, and range.
- Explain the concept of the Vertical Line Test (VLT): If any vertical line intersects the graph at more than one point, then the graph does not represent a function.
- Demonstrate the VLT with several examples using graphs on the slide deck, showing both functions and non-functions.
- Emphasize the 'why' behind the VLT – relating it back to the definition of a function (each input having only one output).
Step 3
Guided Practice: Sorting It Out
15 minutes
- Divide students into small groups or pairs.
- Distribute the cut-out cards from the Vertical Line Test Sort Activity.
- Instruct students to sort the graphs into two categories: 'Functions' and 'Not Functions' using the Vertical Line Test.
- Circulate around the room, providing support and clarification as needed. Encourage groups to discuss their reasoning.
- After 10 minutes, bring the class back together and review a few examples from the sort, having groups share their reasoning.
Step 4
Independent Practice: Graph Analysis Challenge
15 minutes
- Distribute the Graph Analysis Worksheet to each student.
- Students will independently apply the Vertical Line Test to various graphs and determine if they represent a function.
- Encourage students to show their work (e.g., drawing vertical lines on the graphs).
- Circulate to provide individual assistance and check for understanding.
Step 5
Wrap-up & Assessment: Function Check-Up
10 minutes
- Bring the class together for a brief discussion of challenging graphs from the worksheet.
- Distribute the Exit Ticket: Function Check-Up.
- Students will complete the exit ticket independently as a quick assessment of their understanding of the Vertical Line Test.
- Collect exit tickets to gauge comprehension and inform future instruction.
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Slide Deck
Is it a Function?
The Vertical Line Test
Welcome students and introduce the day's topic. Ask them to think about "rules" or "relationships" they encounter every day.
What is a Relation?
- A relation is a set of ordered pairs.
- It shows a relationship between two sets of data (input and output).
- We can represent relations as tables, lists of points, or graphs.
Explain that a relation is simply a set of ordered pairs. Give a few simple examples like (1,2), (2,4), (3,6). Discuss how these can be plotted on a graph.
What is a Function?
- A function is a special type of relation.
- In a function, each input (x-value) has exactly one output (y-value).
- Think: One input, one job!
Introduce the idea that some relations are special. A function has a strict rule: for every single input, there is ONLY one output. Use a vending machine analogy: press 'A1' and you only get 'Coke', not 'Coke' and 'Sprite' at the same time.
Key Vocabulary: Domain & Range
- Domain: All possible input values (x-values) of a relation or function.
- Range: All possible output values (y-values) of a relation or function.
Explain domain as all possible inputs and range as all possible outputs. Give examples related to relations and functions.
The Vertical Line Test
- The Vertical Line Test (VLT) is a quick way to check if a graph represents a function.
- How it works: Imagine drawing vertical lines across the entire graph.
- If any vertical line intersects the graph at MORE THAN ONE point, then the graph IS NOT a function.
- If NO vertical line intersects the graph at more than one point, then the graph IS a function.
Introduce the Vertical Line Test as a visual shortcut. Explain that if a vertical line can touch the graph in more than one place, it means one x-value has multiple y-values, which breaks the function rule.
Example 1: Is This a Function?
(Image of a parabola opening upwards)
Is this a function? Use your Vertical Line Test!
Demonstrate with a graph that IS a function (e.g., a simple parabola or straight line). Draw imagined vertical lines and show they only hit once.
Example 1: Solution
(Image of a parabola with vertical lines drawn, each intersecting at only one point)
YES, this is a function! No vertical line intersects the graph at more than one point.
Reveal the answer and explain why it is a function.
Example 2: Is This a Function?
(Image of a circle)
Is this a function? Use your Vertical Line Test!
Demonstrate with a graph that IS NOT a function (e.g., a circle or a sideways parabola). Draw imagined vertical lines and show they hit twice.
Example 2: Solution
(Image of a circle with vertical lines drawn, some intersecting at two points)
NO, this is NOT a function! Some vertical lines intersect the graph at two points.
Reveal the answer and explain why it is not a function.
Quick Check!
- What is the main rule for a function?
- How do we use the Vertical Line Test?
Provide a quick recap of the main points and transition to the activity.
Activity
Vertical Line Test Sort Activity
Objective: To practice applying the Vertical Line Test to determine if a graph represents a function.
Instructions:
- Cut out each of the graph cards below.
- Work with your partner or small group to apply the Vertical Line Test to each graph.
- Sort the cards into two piles: "Functions" and "Not Functions."
- Be prepared to explain your reasoning for at least two of your sorts to the class.
Graph Cards (Cut Out Each Box)
Card 1
Graph of a straight line with a positive slope (e.g., y = x)
Card 2
Graph of a parabola opening upwards (e.g., y = x^2)
Card 3
Graph of a circle centered at the origin (e.g., x^2 + y^2 = 9)
Card 4
Graph of a vertical line (e.g., x = 3)
Card 5
Graph of a square root function (e.g., y = sqrt(x))
Card 6
Graph of a cubic function (e.g., y = x^3)
Card 7
Graph of a sideways parabola (e.g., x = y^2)
Card 8
Graph of an absolute value function (e.g., y = |x|)
Worksheet
Graph Analysis Worksheet: Is it a Function?
Name: ____________________________
Instructions: For each graph below, use the Vertical Line Test to determine if the graph represents a function. Circle "Function" or "Not a Function" and then explain your reasoning in the space provided.
Graph 1
Graph of a line with a negative slope (e.g., y = -2x + 1)
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Graph 2
Graph of an ellipse (oval shape)
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Graph 3
Graph of a sideways V shape (e.g., x = |y|)
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Graph 4
Graph of a cubic function with an inflection point (e.g., y = x^3 - x)
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Graph 5
Graph of a semi-circle opening downwards (e.g., y = -sqrt(16-x^2))
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Graph 6
Graph of a function with several distinct points that passes the VLT (e.g., a scatter plot where no two points share the same x-coordinate)
Is it a Function? (Circle one): Function / Not a Function
Reasoning:
Quiz
Exit Ticket: Function Check-Up