What you should learn
To write equations to represent relations, given some of the solutions for the equations. NCTM Curriculm Standards 2, 4, 6 - 10
To write equations to represent relations, given some of the solutions for the equations.
NCTM Curriculm Standards 2, 4, 6 - 10
Introduction: As scuba divers descend, the pressure of the water increases. Scuba divers can determine their depth by the pressure. Pressure can be expressed in atmospheres. An atmosphere is equivalent to 14.7 psi (pounds per square inch) of pressure. The table below shows the relationship between atmospheres of pressure and ocean depth.
Suppose we let p represent the atmospheres of pressure, and let d represent the ocean depth. The relation shown in teh table can also be represented by a graph. Graph the points and observe the pattern.
Notice that when the ordered pairs are graphed, they form a linear pattern. You can use a straightedge to draw the line through these points. Points that lie in a linear pattern can be described by an equation. Look at the relationship between the domain and the range to find a pattern that can be described by an equation.
Notice that the differences in the d values are 33 times the differences of the corresponding p values. It seems that this relationship can be described by the equation d=33p. Check to see if this equation is correct by substituting values of p into the equation.
These values do not match those in the first chart. However, note that each corresponding value for d using the equation is 33 more than the d in the relation. Thus, we need to adjust our equation to compensate for this difference. The equation that describes this relation is d = 33p - 33. Since this relation is also a function, we can write this equation as d(p) = 33p - 33.
Exercise 1: Plot the points in the relation shown in the table. Then write an equation in functional notation for the relation.
Since the points form a linear pattern, we know that there is a linear equation that describes this relation.
The differences in the y values are one-fourth the differences in the x values. This pattern implies that he equation may describe the relation. Check the equation for the domain values in teh relation. Check:
The differences in the y values are one-fourth the differences in the x values. This pattern implies that he equation may describe the relation. Check the equation for the domain values in teh relation.
Check:
The equation that represents the relation is . Since this relation is also a function, we can write this equation in functional notation as
Exercise 2: Write an equation in funcitonal notation for the relation graphed below.
Exercise 3: Many times you can generalize the relationships among data you have collected with an equation.
Exploration Spreadsheets In the relation {(-2, 2), (-1, 5), (0, 8), (1, 11), (2, 14)}, the range difference is 3 when the domain difference is 1. You might suggest that the equation of the relation is y = 3x. Let's use a spreadsheet software to check this equation. Each cell of a spreadsheet is named by a letter and number. The letter refers to the column and the number of the row. Enter each x value into cells A1 through A5. Enter the y values into cells B1 through B5. In cell C1, enter the formula A1*3. This formula means take the value in cell A1 and multiply it by 3. Copy this formula to cells C2 through C5. The spreadsheet will automatically change the formula so that the appropriate cell in column A will be used. In cell D1, enter the formula B1 - C1. This formula will subtract the range value of the equation from the the range value of the relation. Copy this formula to cells D2 thorugh D5. Column D tells what number to add to the equation so that it is correct. In this case, the entries in column D are 8's. The correct equation should be y = 3x + 8. YOUR TURN a. What formula would you use in cell C1 to test the first formula found in Example 2? Use a spreadsheet to check all the values in the relation. b. What number would you expect to appear in column D if the equation you found is correct for the given relation?
Exploration Spreadsheets
In the relation {(-2, 2), (-1, 5), (0, 8), (1, 11), (2, 14)}, the range difference is 3 when the domain difference is 1. You might suggest that the equation of the relation is y = 3x. Let's use a spreadsheet software to check this equation. Each cell of a spreadsheet is named by a letter and number. The letter refers to the column and the number of the row. Enter each x value into cells A1 through A5. Enter the y values into cells B1 through B5. In cell C1, enter the formula A1*3. This formula means take the value in cell A1 and multiply it by 3. Copy this formula to cells C2 through C5. The spreadsheet will automatically change the formula so that the appropriate cell in column A will be used. In cell D1, enter the formula B1 - C1. This formula will subtract the range value of the equation from the the range value of the relation. Copy this formula to cells D2 thorugh D5. Column D tells what number to add to the equation so that it is correct. In this case, the entries in column D are 8's. The correct equation should be y = 3x + 8.
In the relation {(-2, 2), (-1, 5), (0, 8), (1, 11), (2, 14)}, the range difference is 3 when the domain difference is 1. You might suggest that the equation of the relation is y = 3x. Let's use a spreadsheet software to check this equation.
Each cell of a spreadsheet is named by a letter and number. The letter refers to the column and the number of the row. Enter each x value into cells A1 through A5. Enter the y values into cells B1 through B5.
In cell C1, enter the formula A1*3. This formula means take the value in cell A1 and multiply it by 3. Copy this formula to cells C2 through C5. The spreadsheet will automatically change the formula so that the appropriate cell in column A will be used.
In cell D1, enter the formula B1 - C1. This formula will subtract the range value of the equation from the the range value of the relation. Copy this formula to cells D2 thorugh D5.
Column D tells what number to add to the equation so that it is correct. In this case, the entries in column D are 8's. The correct equation should be y = 3x + 8.
YOUR TURN
a. What formula would you use in cell C1 to test the first formula found in Example 2? Use a spreadsheet to check all the values in the relation. b. What number would you expect to appear in column D if the equation you found is correct for the given relation?
a. What formula would you use in cell C1 to test the first formula found in Example 2? Use a spreadsheet to check all the values in the relation.
b. What number would you expect to appear in column D if the equation you found is correct for the given relation?
Activity: Pennies are units for measuring the lengths of nails. Nails measured in pennies can range from 2-penny nails to 60 penny nails. The graph below illustrates the lengths of several penny nails.
a. The length of a nail as a function of its penny status can be modeled by a linear function for 2-through 10 penny nails. Write an equation in funcitonal notation for this relationship. b. Find the lengths of a 3 penny nail, a 5 penny nail, and a 10 penny nail.
a. The length of a nail as a function of its penny status can be modeled by a linear function for 2-through 10 penny nails. Write an equation in funcitonal notation for this relationship.
b. Find the lengths of a 3 penny nail, a 5 penny nail, and a 10 penny nail.
Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.
Homework: The homework to be assigned for tonight would be: 11 - 35 odd, 36 - 42
Alternative Homework: Enriched: 12 - 30 even, 31 - 42
Extra Practice: Students book page 768 Lesson 5-6
Extra Practice Worksheet: Click Here.
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