Function or Not? Unmasking Functions in Equations

PreAlgebra Grades High School 3:31 Video
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Lesson Description

Learn to determine if y is a function of x given an equation using graphing, solving for y, and testing values. Explore the vertical line test and real-world applications.

Video Resource

Is y a Function of x?

Mario's Math Tutoring

Duration: 3:31
Watch on YouTube

Key Concepts

  • Definition of a function (one output for every input)
  • Vertical Line Test
  • Solving for y to determine functionality

Learning Objectives

  • Students will be able to define what it means for y to be a function of x.
  • Students will be able to apply the vertical line test to determine if a graph represents a function.
  • Students will be able to algebraically determine if an equation represents y as a function of x by solving for y and analyzing the result.
  • Students will be able to determine if a relation is a function by testing values.

Educator Instructions

  • Introduction (5 mins)
    Begin by defining what it means for y to be a function of x. Emphasize the 'one input, one output' rule. Briefly introduce the vertical line test as a visual method for identifying functions.
  • Vertical Line Test (5 mins)
    Explain the vertical line test in detail. Provide visual examples of graphs that pass and fail the test. Refer to the related video for further clarification.
  • Example 1: Linear Equation (5 mins)
    Work through the first example from the video: y = 3x + 1. Graph the line and demonstrate the vertical line test. Also, show that for any x value plugged into the equation, there is only one corresponding y value.
  • Example 2: y² = 64x (7 mins)
    Address the second example: y² = 64x. Solve for y, emphasizing the importance of considering both positive and negative square roots. Explain why this results in two y-values for a single x-value (e.g., x = 1 yields y = 8 and y = -8). Show the graph as a sideways parabola. Conclude that y is not a function of x in this case.
  • Example 3: 3x² - y = 2 (7 mins)
    Solve 3x² - y = 2 for y. Show the resulting equation, y = 3x² - 2. Explain that this is a parabola. Sketch the graph and apply the vertical line test to confirm that y is a function of x. Note the vertical stretch.
  • Example 4: |y| = 2x (7 mins)
    Begin by explaining that absolute value must be taken into account. Because absolute value returns the positive form of any number, any negative or positive y value returns a positive. Therefore, multiple y's can correspond to any x value. This means it is not a function.
  • Wrap-up and Discussion (4 mins)
    Summarize the different methods for determining if y is a function of x. Open the floor for questions and discussion.

Interactive Exercises

  • Graphing Activity
    Provide students with a list of equations. Have them graph each equation (either by hand or using graphing software) and apply the vertical line test to determine if y is a function of x.
  • Solve and Analyze
    Give students a set of equations and have them solve for y. Then, have them analyze the resulting equation to determine if there's only one possible y-value for each x-value.

Discussion Questions

  • Why is it important to consider both positive and negative roots when solving for y with a squared term?
  • Can you think of a real-world example where a relation is NOT a function? Explain why it fails the 'one input, one output' rule.

Skills Developed

  • Algebraic manipulation
  • Graphical analysis
  • Critical thinking
  • Problem-solving

Multiple Choice Questions

Question 1:

Which of the following best describes a function?

Correct Answer: A relation where each input has exactly one output.

Question 2:

What does the Vertical Line Test determine?

Correct Answer: If a relation is a function

Question 3:

If solving for y results in y = ±√(x+4), is y a function of x?

Correct Answer: No

Question 4:

Which of the following equations represents y as a function of x?

Correct Answer: y = x³ + 1

Question 5:

If a vertical line intersects a graph at three points, what can be concluded?

Correct Answer: The graph does not represent a function.

Question 6:

Which statement is true about the function y = x^2?

Correct Answer: It passes the vertical line test and is a function.

Question 7:

Given x = |y|, is y a function of x?

Correct Answer: No, because for a single value of x, there can be two y values.

Question 8:

What is the first step to determine if y is a function of x when looking at an equation?

Correct Answer: Solve for y.

Question 9:

Which of the following functions will not pass the vertical line test?

Correct Answer: x = y^2

Question 10:

Which type of graph is best to utilize the vertical line test?

Correct Answer: Line graph

Fill in the Blank Questions

Question 1:

For y to be a function of x, each input x must have only ______ output y.

Correct Answer: one

Question 2:

The ______ ______ ______ is a visual test to determine if a graph represents a function.

Correct Answer: vertical line test

Question 3:

If solving for y results in y = ±√x, then y is ______ a function of x.

Correct Answer: not

Question 4:

A relation is NOT a function if one input has _____ than one output.

Correct Answer: more

Question 5:

If a vertical line intersects a graph at more than one point, the relation is _____ a function.

Correct Answer: not

Question 6:

The first step in solving for whether a relation is a function is to attempt to solve for _____

Correct Answer: y

Question 7:

If an equation involves y raised to an even power, one must account for both ________ and ________ values

Correct Answer: positive

Question 8:

A _______ can be used to quickly determine if an equation is a function of x by visualizing it on a graph.

Correct Answer: sketch

Question 9:

The vertical line test works because a line can only intersect a function on one ________.

Correct Answer: point

Question 10:

The relationship between inputs and outputs in a function can be described as one-to-______.

Correct Answer: one

Educational Standards

PC.F.1:Analyze functions and relations. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features.

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