8.F.1-3 Misconceptions

Math Considerations 8.F.1 8.F.2 8.F.3

3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.

A relation is a relationship between sets of information. A function is a special relation. Each element in the first set of objects is paired with one and only one element in the second set of objects. When given a starting point we know exactly where to go; given an x-value, we get one y-value. The starting point is called the “domain” and the ending point is called the “range.”

Many students get confused about what makes a relation a function. They sometimes interchange inputs and outputs, which can lead to an incorrect application of the definition for a function. By practicing with both mathematical and non-mathematical examples, students will increase their understanding of functions and learn to recognize when a relation is a function.

Let’s start with some non-mathematical examples. The pairs of fruit and colors are ordered, meaning one comes first and the other comes second. You can state a color first and then state a matching fruit or vice versa.

RELATION OR FUNCTION? Orange

Orange

Yellow

Lemon

Red

Banana

Green

Strawberry Raspberry

Orange

Orange

Lemon

Yellow

Banana

Red

Strawberry

Green

Raspberry

This is a relation but not a function because some elements in the first set are paired with more than one element (or none) in the second set.

Both red and yellow pair with two fruits and green doesn’t pair with anything.

This is a function because each element in the first set is paired with exactly and only one element in the second set. It doesn’t matter that nothing is paired with green nor does it matter that yellow and red are each paired with two elements.

The domain is {orange, lemon, banana, strawberry, raspberry}.

The range is {orange, yellow, red}. It does not include green because green is not paired with any element in the domain.

Now let’s look at some examples using numbers. The domain values remain the same in each example, but the range values change. Students need to become familiar with vocabulary such as input, output, domain, range, function, and relation to help them be successful in their study of functions.

Domain x -3 -2 -1 0

Range y 3 -6 0 15

1

-1

Input x -3 -2 -1 0 1

Output y -6 -6 -6 -6 -6

Input -3 -2 -1 0 1 Domain -3 -2 -1 0 1 16

This is a function because there is only one range for each domain.

Not sure? There is only one y for each x.

Even though the y is the same for each x, it is still only one y for each x. Therefore, it is a function!

Output -6 -1 0 3 15 Range -6 -1 0 3 15

The 1 in the input is paired with both 0 and 15 in the output, so this is not a function, but it is a relation.

Is this a trick? Each element in the domain has only one partner in the range, except for the 16. It doesn’t have a partner. Remember, each starting point must have one and only one ending point. If the starting point is 16, what is the ending point? There isn’t one. This is not a function, nor is it a relation.

A common “trick” used to test whether something is a function is to use the Vertical Line Test. This test allows us to quickly judge if a graph is a representation of a function. “Each input can have only one output to be a function.” What does this mean for a graph? What do graphs of functions look like? Can they only be straight lines? We know a function must have only one y-value for each x-value. The Vertical Line Test tests this rule. Draw a straight line down any x-value to test whether each x-value has only one y-value.

The illustration below shows several examples of the Vertical Line Test.

• Figures 1, 2, 3, 6, and 8 all pass the test because each x-value has exactly one y-value. They are functions.

• Figures 4, 5, and 7 do not pass the test because each x-value has more than one y-value. Figure 4 is a vertical line and therefore we have only one x-value with an infinite number of y-values. Figures 5 & 7 show two y-values for the same x-value.

WHAT TO DO:

Students who use the Vertical Line Test should be able to explain why it can be used to test whether a graph is a function.

Functional relationships are dependent relationships. This means that the value of one variable is defined in terms of the other variable. For example, we can say, “Profit is a function of the number of T-shirts sold.” The phrase, “is a function of” expresses the dependent relationship. The profit depends on, is a function of, the T-shirt sales.

Listening to how students describe a functional relationship in their own words provides valuable information about their understanding of independent and dependent variables. Expressing a functional relationship in words requires the student to first identify the variables and then to think about which variable is dependent on the other. They have to think about what is changing and how that change is taking place. When students have to put their ideas into words it helps them make sense of what is happening.

SITUATION: Suppose Tanner is a high school student trying to earn money for college. He has an agreement with a local college that during all basketball games and events in the event center he will sell T-shirts. The college charges him $50 per night for the booth rental. Tanner has figured his expenses to purchase the T-shirts and knows that he will make $2.00 profit per T-shirt. Tanner has drawn the graph below to represent his potential profit each night.

WHAT TO DO:

Check to see if students have identified the variable(s) that are changing.

Sometimes discussing what is not changing can help students focus on what is changing.

Functions describe how variables change together.

LINEAR FUNCTION:

In a linear relationship, the ratio of change in input to change in output is always the same. This means, that for any unit increase in the input, the output increases or decreases by a constant amount. This constant change, called the rate of change in a relationship, describes how one variable changes with respect to the other variable. The slope of the line represents the rate of change. The equation has a general form of y = mx + b.

There is a clear pattern to the points on the graph because the profit goes up as the sales go up. The graph shows a relationship between sales and profits in a straight line; it is a linear relationship that is increasing.

Using the graph, we can answer several questions about Tanner’s T-shirt sale profits.

• How many T-shirts must be sold to break even?

• What does it mean when the line crosses the x-axis?

• What is the profit for 60 T-shirts?

• How many T-shirts must be sold to make a profit of $100?

Context gives meaning to the graph, and the graph adds understanding to the context.

The line can be extended indefinitely in both directions. However, it doesn’t make sense to talk about sales that are negative beyond his booth rental, nor does it make sense to talk about sales in the millions.

NONLINEAR RELATIONSHIPS: What if the rate of change is not constant? Can a rate of change be predictable even if it is not constant? Absolutely!

QUADRATIC FUNCTION:

Quadratic functions are characterized by rates of change that are changing at a constant rate. For example, suppose you have a large garden area. You have purchased plants to build a rectangle that has a perimeter of 24 feet.

How many plants should you buy to cover the area of the rectangle you build?

The perimeter is fixed, but the length and width can change which means the area can also change. To make math easy we will work with increments of 1 ft.

If the perimeter is fixed at 24 feet we can write equations for the width and the area as functions of the length. (Remember, in 8th grade students do not use function notation so we will write the functions as equation.)

In order to figure the area, we need to first figure the width.

This graph is linear because it decreases at a constant rate.

Students will gain more insight and practice understanding nonlinear functions in 8.F.5. For now they should just recognize that it is not linear.

EXPONENTIAL FUNCTION:

Exponential functions model situations in which amounts increase or decrease at a rate proportional to the amount given. Like quadratic functions, exponential functions have a rate of change that is not constant.

The volume of a cone (8.G.9) is a related to both the height and the radius. If the height is fixed, the volume is a function of the radius. In the example below, the height is fixed at 6 cm and the radius varies. The resulting graph is a curved line.

r = 6 cm

r = 5 cm

r = 3 cm

r = 1 cm

What type of graph will you get if you use a fixed radius but vary the heights?

Students will gain more insight and practice understanding nonlinear functions in 8.F.5. For now they should just recognize that it is not linear.