Consider the standard form of an exponential function:
y = a(b)^x
1. How does changing the base number (inside the parentheses) change your graph?
2. How does changing the number outside the parentheses change your
graph?
3. Create three new exponential equations to represent Alison, Cindy, and
Javier.
Alison loves social media and likes for her posts to be shared as much as
possible while Cindy and Javier prefer for their posts to stay amongst their
friends primarily
I think I have 1 & 2 figured out
Great! Let's review how changing the base number and the number outside the parentheses affects the graph of an exponential function.
1. How does changing the base number (inside the parentheses) change your graph?
To understand the effect of changing the base number (b) on the graph, we need to consider the following scenarios:
- When the base is greater than 1 (b > 1): Increasing the base amplifies the rate at which the function grows. As x increases, the y-values grow more rapidly, resulting in a steeper graph. On the other hand, decreasing the base (but still greater than 1) makes the function grow more slowly, leading to a flatter graph.
- When the base is between 0 and 1 (0 < b < 1): Increasing the base in this case actually causes the function to shrink. The larger the base, the more the function decreases for each unit increase in x. Conversely, decreasing the base (but still between 0 and 1) makes the function decrease less rapidly, resulting in a less steep graph.
2. How does changing the number outside the parentheses change your graph?
The number outside the parentheses (a) is called the initial value or the y-intercept. Here's what happens when we change this constant:
- Increasing a: If a is increased, it raises the overall level of the graph, shifting it upward. This means that for any given value of x, the corresponding y-value will be higher.
- Decreasing a: When a is decreased, it lowers the entire graph, shifting it downward. This implies that for any given x-value, the corresponding y-value will be lower.
Now, let's move on to the next question:
3. Create three new exponential equations to represent Alison, Cindy, and Javier. Alison loves social media and likes for her posts to be shared as much as possible, while Cindy and Javier prefer for their posts to stay amongst their friends primarily.
We can create exponential equations with different bases and initial values to represent their preferences. Let's assume the variable x represents the number of days:
- Alison: Since she wants her posts to be shared widely, we can choose a base greater than 1 (b > 1) to indicate rapid growth. Let's say Alison's base (b) is 1.5, and her initial value (a) is 100 (meaning she starts with 100 shares). The equation would be:
y = 100(1.5)^x
- Cindy: As Cindy prefers her posts to stay amongst her friends, we can use a base between 0 and 1 (0 < b < 1) to indicate slower growth. Let's say Cindy's base (b) is 0.8, and her initial value (a) is 50 (meaning she starts with 50 shares):
y = 50(0.8)^x
- Javier: If Javier also prefers his posts to stay within his friend circle, we can use similar values as Cindy. Let's say Javier's base (b) is 0.75, and his initial value (a) is 30 (meaning he starts with 30 shares):
y = 30(0.75)^x
These equations represent the growth patterns of Alison, Cindy, and Javier's social media shares over time, based on their preferences.
I hope this explanation helps! If you have any further questions, feel free to ask.