Questions and Answers
What is the recursive function that generates the arithmetic sequence 14, 24, 34, 54,...?
f(n + 1) = f(n) + 10 where f(1) = 14
Which graphs show functions with direct variation? (Check all that apply)
What is the constant variation, k, of the direct variation, y = kx, through (5, 8)?
k = 8/5
Does the function representing bicycle rental costs represent a direct variation? Why?
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Which graph represents a function with direct variation?
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What is the acceleration due to gravity of a falling object based on direct variation with time?
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What can be said about the function through the three graphed points?
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What equation represents Marisa's library fine as a function of the number of days overdue?
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What equation represents the direct variation function containing the points (-8, -6) and (12, 9)?
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What equation represents the direct variation function containing the points (2, 14) and (4, 28)?
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What equation represents the force on an object due to gravity according to its mass?
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What equation describes the direct variation function between total earnings, E, and hours worked, h?
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Which graph represents a function with direct variation and goes through the origin?
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What is the constant of variation for the subscription cost of a newspaper based on the number of weeks?
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Study Notes
Arithmetic Sequences
- An arithmetic sequence is defined by a constant difference between consecutive terms.
- Example: The sequence 14, 24, 34, 54 has a common difference of 10.
- Recursive function for this sequence: f(n + 1) = f(n) + 10 with f(1) = 14.
Direct Variation Functions
- A function with direct variation has a constant ratio (k) between the variables.
- Graphs that show direct variation must pass through the origin; examples include Graph 2, Graph 3, and Graph 5.
- Equation of direct variation: y = kx where k is the constant of variation.
Constant Variation
- The constant of variation (k) represents the constant ratio in direct variation equations.
- For direct variation through the point (5, 8), k is calculated as: k = 8/5.
Cost and Direct Variation
- A function representing costs that passes through the origin indicates direct variation.
- Example: Bicycle rental cost is $5 per hour, which indicates a constant rate of change.
Velocity and Acceleration
- Velocity varies directly with time; this relationship is modeled by the corresponding table.
- The acceleration due to gravity is a direct variation constant measured at 9.8 m/s².
Linear Functions
- A function can be linear but not necessarily represent direct variation, particularly if it does not pass through the origin.
Library Fines and Function Representation
- Marisa's library fines increase in a linear fashion: $0.50, $0.65, $0.80, $0.95, $1.10.
- The function representing her fines as days overdue (n) is: f(n) = 0.15n + 0.35.
Force of Gravity
- The force on an object due to gravity directly varies with its mass.
- The equation representing this relationship is F = 32.2m, where 32.2 ft/s² is the constant of variation.
Earnings from Work
- Direct variation can also relate earnings to hours worked.
- The earnings equation is: E = 7.5h, indicating the total earnings in dollars.
Graphical Representation
- For graphs to be considered direct variation, they must originate from the point (0,0) representing zero value in both variables.
- Notable graphs that demonstrate this include Graph 3.
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Description
Test your knowledge of special linear relationships with this set of flashcards. Engage with various concepts such as arithmetic sequences, direct variation, and recursive functions. Perfect for students looking to strengthen their understanding of mathematics.
