Henrietta did an experiment. She started out with 800 bacteria cells. She found that the growth rate of the bacteria cells was 3.2%. Sketch the graph that represents the situation. Label the y-intercept and the point that represents the projected bacteria population 35 h from the time Henrietta started the experiment.

To represent the situation graphically, we'll follow these steps:

1. **Equation for Exponential Growth:**
   The general formula for exponential growth is:
   \[
   P(t) = P_0 \times (1 + r)^t
   \]
   where:
   - \(P(t)\) is the population at time \(t\).
   - \(P_0\) is the initial population.
   - \(r\) is the growth rate per time period.
   - \(t\) is the time elapsed.

   Given:
   - Initial population (\(P_0\)) = 800 bacteria cells
   - Growth rate (\(r\)) = 3.2% or 0.032
   - Time (\(t\)) is in hours.

   The equation becomes:
   \[
   P(t) = 800 \times (1.032)^t
   \]

2. **Finding the y-intercept:**
   The y-intercept occurs at \(t = 0\):
   \[
   P(0) = 800 \times (1.032)^0 = 800 \text{ cells}
   \]
   So, the y-intercept is at the point \((0, 800)\).

3. **Finding the population at \(t = 35\) hours:**
   Substitute \(t = 35\) into the equation:
   \[
   P(35) = 800 \times (1.032)^{35}
   \]
   I'll calculate this now.The projected bacteria population after 35 hours is approximately 2409 cells. 

So, the point representing the projected bacteria population at \( t = 35 \) hours is \((35, 2409)\).

### Graph Description:
- The y-axis represents the number of bacteria cells.
- The x-axis represents time in hours.
- The graph is an upward-sloping curve (exponential growth).
- The curve starts at the point \((0, 800)\), which is the y-intercept.
- Another key point on the graph is \((35, 2409)\).

Would you like to see a visual representation of the graph, or do you have any other questions? 

### Related Questions:
1. How would the graph change if the growth rate were doubled?
2. What would be the bacteria population after 50 hours?
3. How long would it take for the bacteria population to reach 5000 cells?
4. What would the graph look like if the growth rate were 0%?
5. How would you determine the doubling time for the bacteria population?
6. What effect does decreasing the initial population have on the graph?
7. What if the experiment starts with 1000 bacteria instead of 800?
8. How would you sketch the graph for a negative growth rate?

### Tip:
In exponential growth problems, small changes in the growth rate can lead to large differences in the population over time. Always check the long-term behavior when analyzing such models.

Explore Henrietta's experiment on bacterial growth, starting with 800 cells and a growth rate of 3.2%. Learn to graph the exponential growth curve and identify key points like the y-intercept and the projected population after 35 hours. Suitable for high school students studying exponential functions.

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