1. **Stating the problem:** We start with the exponential growth function $$y=1\cdot 2^x$$ where $x$ is the day number and $y$ is the number of people who know the information. The table shows doubling each day: Day 0 to 5 corresponds to $y=1,2,4,8,16,32$. 2. **Predicting the new growth if Linda tells two people each day and each person tells two others:** This means the growth rate doubles again. The base of the exponential function changes from 2 to 4 because each person tells 2 others, so the number of people quadruples each day. 3. **New equation:** The new function is $$y=1\cdot 4^x$$ where $a=1$ (initial amount) and $b=4$ (growth factor). 4. **Effect of changing $b$:** Increasing $b$ from 2 to 4 makes the function grow faster, so the number of people knowing the information increases more rapidly. 5. **Time to reach at least 500 people:** Solve $$4^x \geq 500$$ Take logarithm base 4: $$x \geq \log_4(500) = \frac{\log(500)}{\log(4)}$$ Calculate approximately: $$x \geq \frac{2.69897}{0.60206} \approx 4.48$$ So it takes about 5 days (since $x$ must be an integer day) to reach at least 500 people. 6. **If on day 0, two people knew instead of one:** The initial amount $a$ changes from 1 to 2, so the equation becomes $$y=2\cdot 4^x$$. 7. **Effect of changing $a$:** Increasing $a$ shifts the graph upward, starting with more people initially knowing the information, but the growth rate $b$ remains the same. **Final answers:** - New equation with doubling each day: $$y=1\cdot 4^x$$ - Time to reach 500 people: about 5 days - New equation if 2 people knew at day 0: $$y=2\cdot 4^x$$