Determine Linear Growth And Decay Worksheet

Algebra 2
Exponential Functions

Total questions - 6

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How Will This Worksheet on "Determine Linear Growth and Decay" Benefit Your Student's Learning?

  • It enhances their ability to predict future values based on current trends, which is crucial for planning and decision-making in various fields like finance and economics.
  • It promotes analytical thinking by requiring students to interpret how quantities change over time, fostering a deeper understanding of mathematical concepts and their real-world applications.
  • Illustrates the sensible use of linear growth and rot models in regular conditions consisting of populace dynamics, monetary planning, and radioactive decay predictions.
  • Strengthens students' analytical capabilities by using requiring them to investigate facts and practice mathematical concepts to clear up troubles concerning increase and rot.
  • Improves the potential to interpret numerical records as it should be and articulate findings simply in both verbal and written formats.

How to Determine Linear Growth and Decay?

  • Determine whether the quantity is increasing (growth) or decreasing (decay) at a constant rate over time.
  • Use the linear equation \( y = mx + b \), where \( m \) represents the rate of change (slope). A positive \( m \) indicates growth, while a negative \( m \) indicates decay.
  • Consider the starting value \( b \) (`y`-intercept) and how the quantity changes over time-based on the rate \( m \).

Solved Example

Q. How does g(x)=10xg(x) = 10^x change over the interval from x=1x = 1 to x=3x = 3?\newlineChoices:\newlineg(x)g(x) increases by a factor of 2020\newlineg(x)g(x) increases by a factor of 100100\newlineg(x)g(x) decreases by 20%20\%\newlineg(x)g(x) decreases by a factor of 1010
Solution:
  1. Evaluate g(x)g(x) at lower bound: Evaluate g(x)g(x) at the lower bound of the interval.\newlineWe need to find the value of g(x)g(x) when x=1x = 1.\newlineCalculate g(1)=101g(1) = 10^1.
  2. Evaluate g(x)g(x) at upper bound: Evaluate g(x)g(x) at the upper bound of the interval.\newlineWe need to find the value of g(x)g(x) when x=3x = 3.\newlineCalculate g(3)=103g(3) = 10^3.
  3. Determine direction of change: Determine the direction of change in g(x)g(x) over the interval.\newlineCompare g(1)g(1) and g(3)g(3) to see if g(x)g(x) increases or decreases.\newlineSince 10310^3 is greater than 10110^1, g(x)g(x) increases from x=1x = 1 to x=3x = 3.
  4. Calculate growth factor: Calculate the factor by which g(x)g(x) increases over the interval.\newlineDivide g(3)g(3) by g(1)g(1) to find the growth factor.\newlineCalculate the growth factor as: \newline103101=1031=102=100\frac{10^3}{10^1} = 10^{3-1} = 10^2 = 100
  5. Match growth factor with choices: Match the calculated growth factor with the given choices.\newlineThe growth factor is 100100, so g(x)g(x) increases by a factor of 100100 from x=1x = 1 to x=3x = 3.
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About Worksheet

Algebra 2
Exponential Functions

To determine linear growth and decay, observe how a quantity changes steadily over time. In growth, the quantity increases by a constant rate per unit of time, depicted as \( y = mx + b \), where \( m \) represents the rate of increase. In contrast, decay involves a steady decrease at a constant rate. Use a worksheet and look at examples to understand how growth and decay are different in real-life situations.

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