Then/Now You recognized arithmetic sequences and related them to linear functions. (Lesson 3–5) Write an equation for a proportional relationship. Write.

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Presentation transcript:

Then/Now You recognized arithmetic sequences and related them to linear functions. (Lesson 3–5) Write an equation for a proportional relationship. Write a relationship for a nonproportional relationship.

Vocabulary inductive reasoning

Concept

Example 1 A Proportional Relationships Graph the data. What can you deduce from the pattern about the relationship between the number of hours driving h and the numbers of miles driven m? A. ENERGY The table shows the number of miles driven for each hour of driving.

Example 1 B Proportional Relationships Look at the relationship between the domain and the range to find a pattern that can be described as an equation. B. Write an equation to describe this relationship.

Example 1 B Proportional Relationships Since this is a linear relationship, the ratio of the range values to the domain values is constant. The difference of the values for h is 1, and the difference of the values for m is 50. This suggests that m = 50h. Check to see if this equation is correct by substituting values of h into the equation.

Example 1 B Proportional Relationships CheckIf h = 1, then m = 50(1) or 50. If h = 2, then m = 50(2) or 100. If h = 3, then m = 50(3) or 150. If h = 4, then m = 50(4) or 200. The equation is correct. Answer: m = 50h

Example 1 B Proportional Relationships C. Use this equation to predict the number of miles driven in 8 hours of driving. m= 50hOriginal equation m= 50(8)Replace h with 8. m= 400 Simplify. Answer: 400 miles

A.A B.B C.C D.D Example 1 CYP A A. Graph the data in the table. What conclusion can you make about the relationship between the number of miles walked and the time spent walking?

A.A B.B C.C D.D Example 1 CYP B

A.A B.B C.C D.D Example 1 CYP C C. Use the equation from part B to predict the number of miles driven in 8 hours.

Example 2 Nonproportional Relationships Write an equation in function notation for the graph.

Example 2 Nonproportional Relationships SolveSelect points from the graph and place them in a table The difference in the x values is 1, and the difference in the y values is 3. The difference in y values is three times the difference of the x values. This suggests that y = 3x. Check this equation.

Example 2 Nonproportional Relationships If x = 1, then y = 3(1) or 3. But the y value for x = 1 is 1. This is a difference of –2. Try some other values in the domain to see if the same difference occurs. y is always 2 less than 3x.

Example 2 Nonproportional Relationships This pattern suggests that 2 should be subtracted from one side of the equation in order to correctly describe the relation. Check y = 3x – 2. If x = 2, then y = 3(2) – 2 or 4. If x = 3, then y = 3(3) – 2 or 7. Answer: y = 3x – 2 correctly describes this relation. Since the relation is also a function, we can write the equation in function notation as f(x) = 3x – 2. Check Compare the ordered pairs from the table to the graph. The points correspond.

A.A B.B C.C D.D Example 2 CYP Write an equation in function notation for the relation that is graphed.