Algebra 2

Graph Analysis: The graph represents the function m, displaying the amount of medicine in the body, measured in milligrams, as a function of time (in hours).
- Estimated Value: To estimate m(0.5):
- Analyze the graph at the point corresponding to 0.5 hours.
Time of Administration: Medicine is administered at noon. To estimate the amount at 4:30 PM:
- Calculate the elapsed time from noon to 4:30 PM, which is 4.5 hours.
- Use the graph to find m(4.5) by locating the corresponding point on the graph.
Decrease Factor Calculation:
  - First Hour: Calculate the factor by which the medicine amount decreased over the first hour (from 0 to 1 hour).
    - Locate m(0) and m(1) on the graph.
    - Compute the decrease factor as:
       $\text{Decrease factor} = \frac{m(0)}{m(1)}$

  - First 1.5 Hours: Calculate the factor of decrease from 0 hours to 1.5 hours.
    - Find m(1.5) on the graph.
    - Compute the decrease factor as:
       $\text{Decrease factor} = \frac{m(0)}{m(1.5)}$

Function Representation: The equation $f(x) = 1.3^x$ represents the number of subscribers (in thousands) an influencer has after tracking for x months.
- Subscriber Metrics: The function suggests exponential growth based on time.
True Statements Validation:
  - Statement g: $\log_3{5000}$ indicates how many months it will take for the influencer to reach 500,000 subscribers.
  - Statement h: The graph suggests that the approximate value of $\log_3{700}$ is around 6.
  - Statement i: $\log_3{y}$ accurately represents a logarithmic form related to the function f(x).
  - Statement j: The influencer is projected to reach 100,000 subscribers shortly after 5 years (60 months).
  - Statement k: Indicates that 5.4 is a reasonable approximation for $\log_3{400}$ based on the graph.

Graph Overview: A graph demonstrating insect population growth over several years, with key coordinates given.
  - Coordinate Points:
    - Point A: (1, 3)
    - Point C: (4, 40)
Growth Factor Calculation:
  - Over 3 Years: Determine the increase factor from year 1 (3 insects) to year 4 (40 insects):
    - Calculate by setting up:
       $\text{Increase Factor} = \frac{40}{3}$

  - Annual Change: Calculate the rate of change over one year (average):
    - Find the population at years 1 and 4, then divide the total increase over three years by 3 to find the approximate annual increase.
    - Establish the annual change as follows:
       $\text{Annual Change} = \frac{40 - 3}{4 - 1}$
Equation Creation:
  - Growth Equation: Develop an equation for population growth per year, which could be modeled as exponential growth. A possible form might be:
    - $P(t) = P_0 e^{rt}$ where:
      - $P(t)$ is the population at time t,
      - $P_0$ is the initial population,
      - $r$ is the growth rate,
      - $t$ is time in years.