Question
Determine whether the equation could represent exponential growth or decay and what is its initial value.
y = 2^x + 7
y = 2^x + 7
Answers
Answered by
GPT-4o mini
To determine whether the equation \( y = 2^x + 7 \) represents exponential growth or decay, we can look at the term \( 2^x \).
- Since \( 2^x \) is an exponential function with a base greater than 1 (specifically, base 2), it represents exponential growth. As \( x \) increases, \( 2^x \) increases rapidly.
Next, we consider the entire function \( y = 2^x + 7 \):
- The +7 shifts the graph of the exponential function \( 2^x \) upwards by 7 units. This does not change the exponential nature of the growth but rather affects the vertical position of the graph.
Now, to find the initial value of the equation, we look at \( y \) when \( x = 0 \):
\[
y = 2^0 + 7 = 1 + 7 = 8
\]
Thus, the initial value of the function is 8.
In conclusion:
- The equation represents **exponential growth**.
- The **initial value** is \( 8 \).
- Since \( 2^x \) is an exponential function with a base greater than 1 (specifically, base 2), it represents exponential growth. As \( x \) increases, \( 2^x \) increases rapidly.
Next, we consider the entire function \( y = 2^x + 7 \):
- The +7 shifts the graph of the exponential function \( 2^x \) upwards by 7 units. This does not change the exponential nature of the growth but rather affects the vertical position of the graph.
Now, to find the initial value of the equation, we look at \( y \) when \( x = 0 \):
\[
y = 2^0 + 7 = 1 + 7 = 8
\]
Thus, the initial value of the function is 8.
In conclusion:
- The equation represents **exponential growth**.
- The **initial value** is \( 8 \).
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